Half-Credit: Convergence Tests and Series in Calculus II

HALF-CREDIT: Sequences and Series

Introduction to Series

A series is the sum of the terms of a sequence. Determining whether a series converges (adds up to a finite value) or diverges (grows without bound) is a central topic in Calculus II. Various tests are used to analyze convergence.

Key Definitions

• Convergent Series: A series whose sequence of partial sums approaches a finite limit.

• Divergent Series: A series whose sequence of partial sums does not approach a finite limit.

• Power Series: An infinite series of the form $\sum_{n=0}^{\infty} a_n (x-c)^n$.

• Taylor Series: A power series representation of a function centered at a point $x=a$.

Convergence Tests for Series

1. Geometric Series Test

• Form: $\sum_{n=0}^{\infty} ar^n$

• Converges if: $|r| < 1$

• Diverges if: $|r| \geq 1$

• Sum: $\frac{a}{1-r}$ if convergent

• Example: $\sum_{n=0}^{\infty} \left(\frac{1}{2}\right)^n$ converges to $2$.

2. Nth Term Test for Divergence

• Form: $\sum a_n$

• Test: If $\lim_{n \to \infty} a_n \neq 0$, the series diverges.

• Note: If $\lim_{n \to \infty} a_n = 0$, the test is inconclusive.

• Example: $\sum_{n=1}^{\infty} \frac{n}{n+1}$ diverges because $\lim_{n \to \infty} \frac{n}{n+1} = 1 \neq 0$.

3. Integral Test

• Form: $\sum_{n=1}^{\infty} a_n$ where $a_n = f(n)$, $f$ is positive, continuous, and decreasing for $n \geq N$.

• Test: If $\int_N^{\infty} f(x) dx$ converges, so does the series. If the integral diverges, so does the series.

• Example: $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges for $p > 1$ (see p-series).

4. p-Series Test

• Form: $\sum_{n=1}^{\infty} \frac{1}{n^p}$

• Converges if: $p > 1$

• Diverges if: $p \leq 1$

• Example: $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges; $\sum_{n=1}^{\infty} \frac{1}{n}$ diverges.

5. Direct Comparison Test

• Compare: $0 \leq a_n \leq b_n$ for all $n$ beyond some index.

• If: $\sum b_n$ converges and $a_n \leq b_n$, then $\sum a_n$ converges.

• If: $\sum a_n$ diverges and $a_n \geq b_n$, then $\sum b_n$ diverges.

6. Limit Comparison Test

• Given: $a_n, b_n > 0$

• Test: If $\lim_{n \to \infty} \frac{a_n}{b_n} = c$ where $0 < c < \infty$, then $\sum a_n$ and $\sum b_n$ both converge or both diverge.

7. Alternating Series Test (Leibniz Test)

• Form: $\sum_{n=1}^{\infty} (-1)^{n-1} b_n$ with $b_n \geq 0$

• Converges if: $b_n$ is decreasing and $\lim_{n \to \infty} b_n = 0$

• Absolute vs. Conditional Convergence: If $\sum |a_n|$ converges, the series is absolutely convergent; otherwise, if only the original series converges, it is conditionally convergent.

8. Ratio Test

• Test: Compute $L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$

• If: $L < 1$, the series converges absolutely.

• If: $L > 1$ or $L = \infty$, the series diverges.

• If: $L = 1$, the test is inconclusive.

9. Root Test

• Test: Compute $L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$

• If: $L < 1$, the series converges absolutely.

• If: $L > 1$ or $L = \infty$, the series diverges.

• If: $L = 1$, the test is inconclusive.

Power Series and Taylor Series

Power Series Representation

• A power series centered at $x=0$ is $\sum_{n=0}^{\infty} a_n x^n$.

• The interval of convergence is the set of $x$ values for which the series converges.

• Example: $\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n$ for $|x| < 1$.

Taylor Series

• The Taylor series for $f(x)$ about $x=a$ is:

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

• Maclaurin series is the Taylor series centered at $a=0$.

• Example: $\sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} x^{2n+1}$

Summary Table: Series Convergence Tests

Test Name	Form/Condition	Converges When	Diverges When
Geometric Series	$\sum ar^n$	$|r| < 1$	$|r| \geq 1$
Nth Term Test	$\lim_{n \to \infty} a_n$	Never proves convergence	If limit $\neq 0$
Integral Test	$a_n = f(n)$, $f$ positive, decreasing	Improper integral converges	Improper integral diverges
p-Series	$\sum \frac{1}{n^p}$	$p > 1$	$p \leq 1$
Direct Comparison	$0 \leq a_n \leq b_n$	$\sum b_n$ converges	$\sum a_n$ diverges
Limit Comparison	$\lim \frac{a_n}{b_n} = c > 0$	Both converge or both diverge	Both converge or both diverge
Alternating Series	$(-1)^{n-1} b_n$, $b_n$ decreasing, $\to 0$	Conditions met	Conditions not met
Ratio Test	$L = \lim \left| \frac{a_{n+1}}{a_n} \right|$	$L < 1$	$L > 1$
Root Test	$L = \lim \sqrt[n]{|a_n|}$	$L < 1$	$L > 1$

Examples of Series from the Assignment

1. $\sum_{n=1}^{\infty} \frac{n!}{2n! + 1}$ Possible Tests: Ratio Test, Nth Term Test

2. $\sum_{n=1}^{\infty} \left( \frac{1}{2^n} - \frac{1}{3^n} \right)$ Possible Tests: Geometric Series Test, Linear Combination of Geometric Series

3. $\sum_{n=2}^{\infty} \frac{1}{1+n^2}$ Possible Tests: Comparison Test, p-Series Test

4. $\sum_{n=2}^{\infty} \frac{1}{n \ln(n)}$ Possible Tests: Integral Test, Comparison Test

5. $\sum_{n=1}^{\infty} \frac{n2^n}{4n^2 + 1}$ Possible Tests: Ratio Test

6. $\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}$ Possible Tests: Alternating Series Test, Absolute/Conditional Convergence

7. $\sum_{n=1}^{\infty} \frac{e^{2n}}{n^n}$ Possible Tests: Root Test, Ratio Test

• Find a power series for $g(x) = \frac{4}{2-x^2}$ centered at 0

  Hint: Express in the form $\frac{A}{1-r}$ and expand as a geometric series.

• Find the Taylor series for $f(x) = \sin(x)$ about $x=0$

  Formula: $\sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} x^{2n+1}$

Summary of Processes and Conditions for Each Test

• Geometric Series Test: Identify $r$; check $|r| < 1$ for convergence.

• Nth Term Test: Compute $\lim_{n \to \infty} a_n$; if not zero, diverges.

• Integral Test: Check if $a_n = f(n)$ is positive, continuous, decreasing; evaluate $\int f(x) dx$ from $N$ to $\infty$.

• p-Series Test: Identify $p$ in $\sum \frac{1}{n^p}$; converges if $p > 1$.

• Direct Comparison Test: Compare $a_n$ to a known series $b_n$; use inequalities to conclude convergence/divergence.

• Limit Comparison Test: Compute $\lim \frac{a_n}{b_n}$; if finite and positive, both series behave the same.

• Alternating Series Test: Check if terms decrease to zero; if so, converges.

• Ratio Test: Compute $L = \lim \left| \frac{a_{n+1}}{a_n} \right|$; $L < 1$ means absolute convergence.

• Root Test: Compute $L = \lim \sqrt[n]{|a_n|}$; $L < 1$ means absolute convergence.

Additional info: The assignment covers all major convergence tests and series representations relevant to Calculus II, including power and Taylor series.