CALCULUS II

Module 3: Infinite Series

This document provides a structured overview of the key concepts, formulas, and theorems for Module 3.

Introduction to Series

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An infinite series is the sum of the terms of an infinite sequence $\{a_n\}$.

  • Sequence: An ordered list of numbers. Ex: $a_n = \frac{1}{n} \rightarrow 1, \frac{1}{2}, \frac{1}{3}, ...$
  • Series: The sum of that list. Ex: $\sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + ...$

The fundamental question is whether the series converges to a finite sum or diverges.

KEY CONCEPT: Sequence of Partial Sums

The convergence of a series is determined by the limit of its sequence of partial sums, $S_n = \sum_{i=1}^{n} a_i$. If $\lim_{n \to \infty} S_n = S$ (a finite value), the series converges to S.

THEOREM: Geometric Series

A series of the form $\sum_{n=1}^{\infty} ar^{n-1} = a + ar + ar^2 + ...$

  • Converges if $|r| < 1$. The sum is given by the formula $S = \frac{a}{1-r}$, where 'a' is the first term.
  • Diverges if $|r| \geq 1$.

THEOREM: The Test for Divergence

If $\lim_{n \to \infty} a_n \neq 0$ or if the limit does not exist, then the series $\sum a_n$ **diverges**.

IMPORTANT NOTE

The converse of the Test for Divergence is false. If $\lim_{n \to \infty} a_n = 0$, the test is inconclusive. The series may converge or diverge. For example, for the Harmonic Series $\sum \frac{1}{n}$, the terms approach zero, but the series diverges.

Other Convergence Tests

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For series where the Test for Divergence is inconclusive, the following tests can determine convergence.

Guidelines for Selecting a Test

  1. If $a_n = f(n)$ and $\int_1^\infty f(x)dx$ is easily evaluated $\rightarrow$ **Integral Test**.
  2. If the series is of the form $\sum \frac{1}{n^p}$ $\rightarrow$ **p-Series Test**.
  3. If $a_n$ is a rational or algebraic function of n, compare to a p-series $\rightarrow$ **Comparison or Limit Comparison Test**.
  4. If the series terms alternate in sign $\rightarrow$ **Alternating Series Test**.
  5. If the series involves factorials or n-th powers $\rightarrow$ **Ratio or Root Test**.
  • The Integral Test: If $f(x)$ is a positive, continuous, and decreasing function for $x \ge 1$ and $a_n=f(n)$, then $\sum a_n$ and $\int_1^\infty f(x)dx$ either both converge or both diverge.
  • The p-Series Test: The series $\sum \frac{1}{n^p}$ converges if $p > 1$ and diverges if $p \leq 1$.
  • The Direct Comparison Test: Given $0 \le a_n \le b_n$, if $\sum b_n$ converges, then $\sum a_n$ converges. If $\sum a_n$ diverges, then $\sum b_n$ diverges.
  • The Limit Comparison Test: If $\lim_{n \to \infty} \frac{a_n}{b_n} = L$ where L is a finite, positive number, then $\sum a_n$ and $\sum b_n$ either both converge or both diverge.
  • The Alternating Series Test: For $\sum (-1)^n b_n$, if $b_{n+1} \le b_n$ (decreasing) and $\lim_{n \to \infty} b_n = 0$, the series converges.
  • The Ratio Test: Let $L = \lim_{n \to \infty} |\frac{a_{n+1}}{a_n}|$. If $L<1$, the series is absolutely convergent. If $L>1$, the series is divergent. If $L=1$, the test is inconclusive.
  • The Root Test: Let $L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$. The conclusions are the same as for the Ratio Test.

IMPORTANT NOTE

For the Ratio and Root tests, a result of $L=1$ is inconclusive. Another test must be applied.

Power Series

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A power series is a series of the form $\sum_{n=0}^{\infty} c_n (x-a)^n$, where $a$ is the center. It can be viewed as a polynomial of infinite degree.

Objective: Find Convergence Set

For any power series, the primary goal is to find its **Radius of Convergence (R)** and **Interval of Convergence (I.O.C.)**.

STEP 1: Apply the Ratio Test to $\sum |c_n (x-a)^n|$ to find the condition on $x$ for which the series converges. This yields the radius R.

STEP 2: The series converges absolutely for $|x-a| < R$. This defines an open interval $(a-R, a+R)$.

STEP 3: CRITICAL! The convergence at the endpoints, $x = a-R$ and $x = a+R$, must be tested separately by substituting these values into the original series and applying an appropriate convergence test from the previous section.

Representing Functions as Power Series

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Many functions can be represented by a power series. This can be achieved by manipulating known series, primarily the geometric series.

Fundamental Series

The geometric series formula is the foundation for this topic: $\frac{1}{1-x} = \sum_{n=0}^\infty x^n$, which converges for $|x|<1$.

  • Substitution: Functions can be manipulated to fit the form $\frac{a}{1-r}$. For example, to find the series for $\frac{1}{1+x^2}$, rewrite it as $\frac{1}{1-(-x^2)}$ and substitute $-x^2$ for $x$ in the geometric series formula.
  • Differentiation and Integration: Within its interval of convergence, a power series can be differentiated or integrated term-by-term to produce a new power series representing the derivative or integral of the original function. The radius of convergence remains unchanged, but endpoint convergence may change.

Taylor and Maclaurin Series

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The Taylor series provides a general method for representing a function as a power series, provided the function has derivatives of all orders.

DEFINITION: Taylor Series

The Taylor series for a function $f(x)$ centered at $x=a$ is defined as:

$$ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n $$

This expands to: $f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...$

A **Maclaurin Series** is a special case of the Taylor Series, centered at $a=0$.

Common Maclaurin Series Expansions

Committing these essential series to memory is highly efficient for problem-solving.

  • $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...$ (R = $\infty$)
  • $\sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - ...$ (R = $\infty$)
  • $\cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - ...$ (R = $\infty$)
  • $\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + ...$ (R = 1)
  • $\ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1} x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - ...$ (R = 1)

IMPORTANT NOTE

When applying the Taylor series formula, ensure accuracy in calculating the derivatives $f^{(n)}(a)$ at the center point 'a', and do not omit the $n!$ term in the denominator.

Concluding Remarks

A systematic approach is key to mastering this material. Review each section, focusing on understanding the conditions for each theorem and practicing a variety of problems. Consistent effort will lead to success.