Infinite series and sequences are fundamental concepts in calculus and analysis, offering powerful tools for approximating functions and understanding convergence. At conduct.edu.vn, we aim to provide students with a clear and comprehensive guide to mastering these essential mathematical topics. This guide offers a detailed exploration of infinite sequences and series, providing a solid foundation for success in calculus and beyond. Dive in to elevate your understanding of mathematical analysis and convergent sequences.
1. Introduction to Sequences
1.1 What is a Sequence?
In mathematics, a sequence is an ordered list of elements, often numbers, that follow a specific pattern or rule. Understanding sequences is fundamental in mathematical analysis as it lays the groundwork for more advanced concepts like series and limits. A sequence can be finite or infinite, but in the context of calculus and analysis, we are primarily concerned with infinite sequences.
Formal Definition: An infinite sequence is a function whose domain is the set of natural numbers (1, 2, 3, …). The sequence is typically denoted as {a_n}, where ‘a_n’ represents the nth term of the sequence.
Examples of Sequences:
- Arithmetic Sequence: A sequence where the difference between consecutive terms is constant. For example, {2, 4, 6, 8, …} where a_n = 2n.
- Geometric Sequence: A sequence where the ratio between consecutive terms is constant. For example, {1, 2, 4, 8, …} where a_n = 2^(n-1).
- Harmonic Sequence: A sequence where the terms are the reciprocals of the natural numbers. For example, {1, 1/2, 1/3, 1/4, …} where a_n = 1/n.
- Fibonacci Sequence: A sequence where each term is the sum of the two preceding terms. For example, {1, 1, 2, 3, 5, 8, …} defined recursively as an = a(n-1) + a_(n-2) with a_1 = 1 and a_2 = 1.
Understanding the different types of sequences and their underlying patterns is crucial for predicting the behavior of the sequence as n approaches infinity. These patterns can be expressed through explicit formulas or recursive definitions, each providing a unique way to define and analyze sequences.
1.2 Convergence and Divergence of Sequences
A fundamental aspect of analyzing sequences is determining their behavior as n approaches infinity. A sequence is said to either converge, meaning its terms approach a specific value, or diverge, meaning its terms do not approach a specific value.
Convergence: A sequence {a_n} converges to a limit L if, for every ε > 0, there exists an integer N such that for all n > N, |a_n – L| < ε. This means that the terms of the sequence get arbitrarily close to L as n becomes large. Mathematically, this is expressed as:
lim (n→∞) a_n = LDivergence: A sequence {a_n} diverges if it does not converge to any finite limit. Divergence can occur in several ways:
- The terms of the sequence increase or decrease without bound (approach infinity or negative infinity).
- The terms oscillate between two or more values.
- The terms behave erratically without settling down to a specific value.
Examples of Convergence and Divergence:
- Convergent Sequence: The sequence {1/n} converges to 0. For any ε > 0, we can find an N such that for all n > N, |1/n – 0| < ε.
- Divergent Sequence (Unbounded): The sequence {n} diverges to infinity. As n increases, the terms of the sequence also increase without bound.
- Divergent Sequence (Oscillating): The sequence {(-1)^n} diverges because it oscillates between -1 and 1, never approaching a single limit.
Techniques for Determining Convergence:
- Limit Laws: Applying algebraic properties of limits to simplify the sequence.
- Squeeze Theorem: If a_n ≤ b_n ≤ c_n for all n beyond a certain point, and both a_n and c_n converge to L, then b_n also converges to L.
- Monotone Convergence Theorem: A bounded, monotone (either increasing or decreasing) sequence always converges.
Understanding the convergence and divergence of sequences is critical for the study of infinite series. The behavior of a sequence determines whether its corresponding series converges or diverges, making this concept essential for calculus and real analysis. Analyzing these behaviors often requires a blend of theoretical understanding and practical techniques, ensuring a solid grasp of the underlying principles.
1.3 Common Types of Sequences and Their Properties
Understanding the characteristics of different types of sequences helps in predicting their behavior and applying appropriate analytical techniques. Here’s an overview of several common types of sequences:
1. Arithmetic Sequences
- Definition: A sequence in which the difference between any two consecutive terms is constant.
- General Form: a_n = a_1 + (n – 1)d, where a_1 is the first term and d is the common difference.
- Properties:
- Linear growth or decay.
- The nth term can be easily calculated if a_1 and d are known.
- Example: {2, 5, 8, 11, …} with a_1 = 2 and d = 3.
2. Geometric Sequences
- Definition: A sequence in which the ratio between any two consecutive terms is constant.
- General Form: a_n = a_1 * r^(n – 1), where a_1 is the first term and r is the common ratio.
- Properties:
- Exponential growth or decay.
- Convergence depends on the value of r: converges to 0 if |r| < 1, converges to a_1 if r = 1, diverges otherwise.
- Example: {3, 6, 12, 24, …} with a_1 = 3 and r = 2.
3. Harmonic Sequences
- Definition: A sequence formed by taking the reciprocals of an arithmetic sequence.
- General Form: a_n = 1 / (a_1 + (n – 1)d), where a_1 + (n – 1)d is an arithmetic sequence.
- Properties:
- Terms approach 0 as n approaches infinity.
- The harmonic series (sum of the terms of the harmonic sequence) diverges.
- Example: {1, 1/2, 1/3, 1/4, …} which is the reciprocal of the arithmetic sequence {1, 2, 3, 4, …}.
4. Fibonacci Sequence
- Definition: A sequence in which each term is the sum of the two preceding terms.
- Recursive Form: an = a(n-1) + a_(n-2), with a_1 = 1 and a_2 = 1.
- Properties:
- Appears in various mathematical and natural phenomena.
- The ratio of consecutive Fibonacci numbers converges to the golden ratio φ ≈ 1.618.
- Example: {1, 1, 2, 3, 5, 8, 13, …}.
5. Alternating Sequences
- Definition: A sequence in which the terms alternate in sign.
- General Form: a_n = (-1)^n b_n or a_n = (-1)^(n+1) b_n, where b_n is a positive sequence.
- Properties:
- Convergence depends on the behavior of b_n.
- If b_n converges to 0, the alternating sequence converges to 0.
- Example: {-1, 2, -3, 4, -5, …} where a_n = (-1)^n * n.
Table Summarizing Sequence Types and Properties
| Sequence Type | Definition | General Form | Convergence Properties |
|---|---|---|---|
| Arithmetic | Constant difference between consecutive terms | a_n = a_1 + (n – 1)d | Diverges unless d = 0 |
| Geometric | Constant ratio between consecutive terms | a_n = a_1 * r^(n – 1) | Converges to 0 if |r| < 1, converges to a_1 if r = 1, diverges otherwise |
| Harmonic | Reciprocals of an arithmetic sequence | a_n = 1 / (a_1 + (n – 1)d) | Converges to 0 |
| Fibonacci | Each term is the sum of the two preceding terms | an = a(n-1) + a_(n-2) | Diverges |
| Alternating | Terms alternate in sign | a_n = (-1)^n * b_n | Converges to 0 if b_n converges to 0 |
Understanding these sequence types and their properties enables students to quickly identify patterns and apply relevant techniques for convergence analysis. Each type has unique characteristics that make it essential in various mathematical applications.
2. Introduction to Infinite Series
2.1 What is an Infinite Series?
An infinite series is the sum of an infinite sequence of terms. Understanding infinite series is crucial in calculus and mathematical analysis because it allows us to represent functions and solve problems that are otherwise intractable. An infinite series is denoted as:
∑ (n=1 to ∞) a_n = a_1 + a_2 + a_3 + ...Here, a_n is the nth term of the sequence being summed. Unlike finite sums, infinite series may or may not converge to a finite value. The concept of convergence is central to the study of infinite series.
Examples of Infinite Series:
- Geometric Series: 1 + 1/2 + 1/4 + 1/8 + …
- Harmonic Series: 1 + 1/2 + 1/3 + 1/4 + …
- Alternating Harmonic Series: 1 – 1/2 + 1/3 – 1/4 + …
- Power Series: ∑ (n=0 to ∞) c_n * (x – a)^n, where c_n are coefficients and a is a constant.
Partial Sums:
To analyze an infinite series, we often consider its partial sums. The nth partial sum, S_n, of the series ∑ (n=1 to ∞) a_n is the sum of the first n terms:
S_n = a_1 + a_2 + ... + a_nThe sequence of partial sums {S_n} is used to determine whether the series converges or diverges.
Convergence and Divergence:
An infinite series ∑ (n=1 to ∞) a_n converges if the sequence of its partial sums {S_n} converges to a finite limit L. In this case, we write:
∑ (n=1 to ∞) a_n = LIf the sequence of partial sums {S_n} diverges (does not approach a finite limit), then the infinite series diverges. Divergence can occur in several ways, including the partial sums increasing without bound or oscillating.
Understanding these basic concepts is essential before delving into the various tests and techniques used to determine the convergence or divergence of infinite series. Knowing the definition of an infinite series and how to analyze its partial sums provides a foundation for more complex analysis.
2.2 Convergence and Divergence of Infinite Series
The convergence or divergence of an infinite series is a critical concept in mathematical analysis. It determines whether the sum of an infinite number of terms approaches a finite value or not.
Convergence:
An infinite series ∑ (n=1 to ∞) a_n converges if the sequence of its partial sums {S_n} converges to a finite limit L. Mathematically, this means:
lim (n→∞) S_n = LWhere:
- S_n = a_1 + a_2 + … + a_n is the nth partial sum of the series.
- L is the limit to which the partial sums converge.
In this case, we say that the sum of the infinite series is L, and we write:
∑ (n=1 to ∞) a_n = LDivergence:
An infinite series ∑ (n=1 to ∞) a_n diverges if the sequence of its partial sums {S_n} does not converge to a finite limit. Divergence can occur in several ways:
- Divergence to Infinity: The partial sums increase without bound, i.e., lim (n→∞) S_n = ∞.
- Oscillatory Divergence: The partial sums oscillate between two or more values, never settling down to a single limit.
- Erratic Divergence: The partial sums behave irregularly without approaching any specific value.
Examples of Convergence and Divergence:
- Convergent Series (Geometric Series):
- The series 1 + 1/2 + 1/4 + 1/8 + … converges to 2.
- The nth partial sum is S_n = 2 – (1/2)^(n-1), and lim (n→∞) S_n = 2.
- Divergent Series (Harmonic Series):
- The series 1 + 1/2 + 1/3 + 1/4 + … diverges to infinity.
- The partial sums S_n increase without bound as n increases.
- Divergent Series (Oscillating Series):
- The series 1 – 1 + 1 – 1 + … diverges because the partial sums oscillate between 1 and 0.
Key Tests for Convergence and Divergence:
- Divergence Test (nth-Term Test): If lim (n→∞) a_n ≠ 0, then the series ∑ (n=1 to ∞) a_n diverges. This test is only for divergence; it cannot prove convergence.
- Geometric Series Test: A geometric series ∑ (n=1 to ∞) a*r^(n-1) converges if |r| < 1 and diverges if |r| ≥ 1. If it converges, the sum is a / (1 – r).
- Integral Test: If f(x) is a continuous, positive, and decreasing function on [1, ∞), then the series ∑ (n=1 to ∞) f(n) and the integral ∫ (1 to ∞) f(x) dx either both converge or both diverge.
Understanding these concepts and tests is fundamental for determining the behavior of infinite series. The ability to differentiate between convergent and divergent series is crucial in many areas of mathematics and physics, making this a key topic for students to master.
2.3 Tests for Convergence and Divergence
Determining whether an infinite series converges or diverges often requires the use of specific tests. Here, we outline some of the most common and effective tests used in mathematical analysis.
1. The Divergence Test (nth-Term Test)
- Statement: If lim (n→∞) a_n ≠ 0, then the series ∑ (n=1 to ∞) a_n diverges.
- Explanation: This test states that if the terms of the sequence do not approach zero, the series cannot converge.
- Example: For the series ∑ (n=1 to ∞) n/(n+1), lim (n→∞) n/(n+1) = 1 ≠ 0, so the series diverges.
- Note: This test can only be used to prove divergence. If lim (n→∞) a_n = 0, the test is inconclusive, and another test must be used.
2. The Geometric Series Test
- Statement: A geometric series ∑ (n=0 to ∞) a*r^n converges if |r| < 1 and diverges if |r| ≥ 1. If it converges, the sum is a / (1 – r).
- Explanation: Geometric series have a constant ratio between terms, making their convergence behavior predictable.
- Example: The series ∑ (n=0 to ∞) (1/2)^n converges because |1/2| < 1, and its sum is 1 / (1 – 1/2) = 2. The series ∑ (n=0 to ∞) 2^n diverges because |2| ≥ 1.
3. The Integral Test
- Statement: If f(x) is a continuous, positive, and decreasing function on [1, ∞), then the series ∑ (n=1 to ∞) f(n) and the integral ∫ (1 to ∞) f(x) dx either both converge or both diverge.
- Explanation: This test relates the convergence of a series to the convergence of an improper integral.
- Example: For the series ∑ (n=1 to ∞) 1/n^2, the function f(x) = 1/x^2 is continuous, positive, and decreasing on [1, ∞). The integral ∫ (1 to ∞) 1/x^2 dx converges, so the series ∑ (n=1 to ∞) 1/n^2 also converges.
4. The Comparison Test
- Statement: Suppose ∑ a_n and ∑ b_n are series with positive terms.
- If ∑ b_n converges and a_n ≤ b_n for all n, then ∑ a_n also converges.
- If ∑ b_n diverges and a_n ≥ b_n for all n, then ∑ a_n also diverges.
- Explanation: This test compares a given series to a known convergent or divergent series.
- Example: For the series ∑ (n=1 to ∞) 1/(n^2 + 1), we can compare it to the convergent series ∑ (n=1 to ∞) 1/n^2. Since 1/(n^2 + 1) ≤ 1/n^2 for all n, the series ∑ (n=1 to ∞) 1/(n^2 + 1) also converges.
5. The Limit Comparison Test
- Statement: Suppose ∑ a_n and ∑ b_n are series with positive terms. If lim (n→∞) (a_n / b_n) = c, where 0 < c < ∞, then either both series converge or both series diverge.
- Explanation: This test compares the limiting behavior of two series to determine convergence.
- Example: For the series ∑ (n=1 to ∞) (n + 1)/(n^3 + 2), we can compare it to the convergent series ∑ (n=1 to ∞) 1/n^2. Since lim (n→∞) [(n + 1)/(n^3 + 2)] / [1/n^2] = 1, both series converge.
6. The Ratio Test
- Statement: For a series ∑ an, let L = lim (n→∞) |a(n+1) / a_n|.
- If L < 1, the series converges absolutely.
- If L > 1 or L = ∞, the series diverges.
- If L = 1, the test is inconclusive.
- Explanation: This test is particularly useful for series involving factorials or exponential terms.
- Example: For the series ∑ (n=1 to ∞) n/3^n, L = lim (n→∞) [(n+1)/3^(n+1)] / [n/3^n] = 1/3 < 1, so the series converges absolutely.
7. The Root Test
- Statement: For a series ∑ a_n, let L = lim (n→∞) |a_n|^(1/n).
- If L < 1, the series converges absolutely.
- If L > 1 or L = ∞, the series diverges.
- If L = 1, the test is inconclusive.
- Explanation: This test is useful when terms involve nth powers.
- Example: For the series ∑ (n=1 to ∞) (2n/(n+1))^n, L = lim (n→∞) |(2n/(n+1))^n|^(1/n) = 2 < 1, so the series converges absolutely.
Table Summarizing Convergence and Divergence Tests
| Test | Statement | Use Case |
|---|---|---|
| Divergence Test | If lim (n→∞) a_n ≠ 0, then ∑ a_n diverges. | Proving divergence when terms do not approach zero. |
| Geometric Series Test | ∑ a*r^n converges if | r |
| Integral Test | If f(x) is continuous, positive, and decreasing, ∑ f(n) and ∫ f(x) dx either both converge or both diverge. | Series that can be related to an improper integral. |
| Comparison Test | If ∑ b_n converges and a_n ≤ b_n, then ∑ a_n converges. If ∑ b_n diverges and a_n ≥ b_n, then ∑ a_n diverges. | Comparing with a known convergent or divergent series. |
| Limit Comparison Test | If lim (n→∞) (a_n / b_n) = c, where 0 < c < ∞, then ∑ a_n and ∑ b_n either both converge or both diverge. | Comparing with a series with similar limiting behavior. |
| Ratio Test | Let L = lim (n→∞) | a_(n+1) / a_n |
| Root Test | Let L = lim (n→∞) | a_n |
By mastering these tests, students can effectively determine the convergence or divergence of a wide variety of infinite series. Each test has its strengths and weaknesses, and choosing the appropriate test is a critical skill in mathematical analysis.
3. Special Types of Series
3.1 Geometric Series
Geometric series are a fundamental type of infinite series characterized by a constant ratio between consecutive terms. Understanding their convergence and sum is essential in various fields, including calculus, physics, and economics.
Definition:
A geometric series is an infinite series of the form:
∑ (n=0 to ∞) a*r^n = a + ar + ar^2 + ar^3 + ...Where:
- a is the first term of the series.
- r is the common ratio between consecutive terms.
Convergence:
The convergence of a geometric series depends entirely on the value of the common ratio r. The series converges if the absolute value of r is less than 1, i.e., |r| < 1. If |r| ≥ 1, the series diverges.
Sum of a Convergent Geometric Series:
When a geometric series converges (|r| < 1), its sum can be calculated using the formula:
S = a / (1 - r)Examples:
-
Convergent Geometric Series:
- Consider the series: 1 + 1/2 + 1/4 + 1/8 + …
- Here, a = 1 and r = 1/2. Since |1/2| < 1, the series converges.
- The sum of the series is: S = 1 / (1 – 1/2) = 1 / (1/2) = 2.
-
Divergent Geometric Series:
- Consider the series: 1 + 2 + 4 + 8 + …
- Here, a = 1 and r = 2. Since |2| ≥ 1, the series diverges.
- The partial sums increase without bound, and the series does not approach a finite limit.
-
Geometric Series with Alternating Signs:
- Consider the series: 1 – 1/3 + 1/9 – 1/27 + …
- Here, a = 1 and r = -1/3. Since |-1/3| < 1, the series converges.
- The sum of the series is: S = 1 / (1 – (-1/3)) = 1 / (4/3) = 3/4.
Applications:
Geometric series have numerous applications in various fields:
- Calculus: Representing functions as power series.
- Physics: Modeling decay processes, such as radioactive decay.
- Economics: Calculating the present value of an annuity or perpetuity.
Understanding the convergence and sum of geometric series is a fundamental concept in mathematical analysis. The ability to quickly identify a geometric series and apply the appropriate formula is a valuable skill for students and professionals alike.
3.2 Harmonic Series and p-Series
Harmonic and p-series are essential in understanding the convergence and divergence of infinite series. They provide critical examples and serve as benchmarks for comparison tests.
Harmonic Series:
-
Definition: The harmonic series is an infinite series of the form:
∑ (n=1 to ∞) 1/n = 1 + 1/2 + 1/3 + 1/4 + ... -
Divergence: The harmonic series is a classic example of a divergent series. Although the terms 1/n approach zero as n approaches infinity, the sum of the series does not converge to a finite value.
-
Proof of Divergence:
-
One way to prove the divergence of the harmonic series is by grouping terms:
1 + 1/2 + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + …
1 + 1/2 + (1/4 + 1/4) + (1/8 + 1/8 + 1/8 + 1/8) + …
= 1 + 1/2 + 1/2 + 1/2 + … -
Since we can group the terms in such a way that the sum is greater than an infinite sum of 1/2, the harmonic series diverges.
-
-
Applications:
- The harmonic series serves as a critical benchmark for comparison tests. Many series are compared to the harmonic series to determine their convergence or divergence.
p-Series:
-
Definition: A p-series is an infinite series of the form:
∑ (n=1 to ∞) 1/n^p = 1 + 1/2^p + 1/3^p + 1/4^p + ...Where p is a positive real number.
-
Convergence: The convergence of a p-series depends on the value of p:
- If p > 1, the p-series converges.
- If p ≤ 1, the p-series diverges.
-
Examples:
- p > 1 (Convergent):
- Consider the series: ∑ (n=1 to ∞) 1/n^2 = 1 + 1/4 + 1/9 + 1/16 + …
- Here, p = 2 > 1, so the series converges.
- The sum of this series is known to be π^2 / 6.
- p = 1 (Divergent – Harmonic Series):
- Consider the series: ∑ (n=1 to ∞) 1/n = 1 + 1/2 + 1/3 + 1/4 + …
- Here, p = 1, so the series diverges (as discussed above).
- p < 1 (Divergent):
- Consider the series: ∑ (n=1 to ∞) 1/√n = 1 + 1/√2 + 1/√3 + 1/√4 + …
- Here, p = 1/2 < 1, so the series diverges.
- p > 1 (Convergent):
-
Integral Test Proof:
- The convergence of the p-series can be formally proven using the integral test. Consider the function f(x) = 1/x^p.
- The integral ∫ (1 to ∞) 1/x^p dx converges if p > 1 and diverges if p ≤ 1. Therefore, the series ∑ (n=1 to ∞) 1/n^p converges if p > 1 and diverges if p ≤ 1.
Table Summarizing Harmonic and p-Series Convergence
| Series Type | Definition | Convergence Condition | Convergence/Divergence |
|---|---|---|---|
| Harmonic | ∑ (n=1 to ∞) 1/n | p = 1 | Diverges |
| p-Series | ∑ (n=1 to ∞) 1/n^p | p > 1 | Converges |
| p ≤ 1 | Diverges |
Understanding the harmonic and p-series is critical for mastering the concepts of convergence and divergence. These series provide essential benchmarks for comparison tests and are fundamental in advanced calculus and real analysis.
3.3 Alternating Series
Alternating series are a special type of infinite series where the terms alternate in sign. These series have unique convergence properties, and the Alternating Series Test provides a straightforward way to determine their convergence.
Definition:
An alternating series is an infinite series whose terms alternate in sign. It can be written in one of the following forms:
∑ (n=1 to ∞) (-1)^(n-1) * b_n = b_1 - b_2 + b_3 - b_4 + ...
∑ (n=1 to ∞) (-1)^n * b_n = -b_1 + b_2 - b_3 + b_4 - ...Where b_n > 0 for all n.
The Alternating Series Test (Leibniz’s Test):
The Alternating Series Test states that an alternating series converges if the following two conditions are met:
- The sequence {b_n} is decreasing:
- b_(n+1) ≤ b_n for all n.
- The sequence {b_n} converges to 0:
- lim (n→∞) b_n = 0
If these conditions are satisfied, the alternating series converges.
Examples:
-
Convergent Alternating Series (Alternating Harmonic Series):
- Consider the series: 1 – 1/2 + 1/3 – 1/4 + …
- This is an alternating series with b_n = 1/n.
- The sequence {1/n} is decreasing since 1/(n+1) < 1/n for all n.
- The sequence {1/n} converges to 0 since lim (n→∞) 1/n = 0.
- Therefore, by the Alternating Series Test, the series 1 – 1/2 + 1/3 – 1/4 + … converges.
-
Divergent Alternating Series:
- Consider the series: ∑ (n=1 to ∞) (-1)^(n-1) * (n/(n+1)) = 1/2 – 2/3 + 3/4 – 4/5 + …
- This is an alternating series with b_n = n/(n+1).
- The sequence {n/(n+1)} converges to 1, not 0.
- Therefore, by the Alternating Series Test (specifically, the condition that b_n must converge to 0 is not met), the series diverges.
-
Estimating the Sum of a Convergent Alternating Series:
- One useful property of convergent alternating series is that the error in approximating the sum S by the nth partial sum S_n is no greater than the absolute value of the (n+1)th term, i.e., |S – Sn| ≤ b(n+1).
- For example, to approximate the sum of the alternating harmonic series 1 – 1/2 + 1/3 – 1/4 + … with an error less than 0.1, we need to find an n such that 1/(n+1) ≤ 0.1. This gives n ≥ 9. So, summing the first 9 terms will give an approximation within 0.1 of the true sum.
Table Summarizing Key Aspects of Alternating Series
| Aspect | Description |
|---|---|
| Definition | Series with terms that alternate in sign |
| General Form | ∑ (n=1 to ∞) (-1)^(n-1) b_n or ∑ (n=1 to ∞) (-1)^n b_n, where b_n > 0 |
| Alternating Series Test | Converges if {b_n} is decreasing and lim (n→∞) b_n = 0 |
| Error Estimation | |S – Sn| ≤ b(n+1) |
Understanding alternating series and the Alternating Series Test is essential for analyzing the convergence of many infinite series. The error estimation property also provides a practical tool for approximating the sum of these series to a desired level of accuracy.
4. Power Series
4.1 Definition and Properties of Power Series
Power series are a critical concept in mathematical analysis, providing a way to represent functions as infinite sums of powers. This representation is highly valuable in calculus, differential equations, and complex analysis.
Definition:
A power series is an infinite series of the form:
∑ (n=0 to ∞) c_n * (x - a)^n = c_0 + c_1(x - a) + c_2(x - a)^2 + c_3(x - a)^3 + ...Where:
- x is a variable.
- c_n are the coefficients of the series.
- a is a constant called the center of the series.
Key Properties:
- Convergence:
- A power series may converge for some values of x and diverge for others.
- The set of values for which the series
