Series Convergence and Partial Sums in Calculus II

Series and Convergence

Partial Sums and Series Notation

In calculus, a series is the sum of the terms of a sequence. The partial sum $S_n$ is the sum of the first $n$ terms of a series. Determining whether a series converges or diverges is a central topic in Calculus II.

• General Series Notation: $\sum_{n=1}^{\infty} a_n$

• Partial Sum: $S_n = a_1 + a_2 + \cdots + a_n$

• Convergence: A series converges if $\lim_{n \to \infty} S_n$ exists and is finite.

• Divergence: A series diverges if $\lim_{n \to \infty} S_n$ does not exist or is infinite.

Example: Logarithmic Series

Consider the series $\sum_{n=1}^{\infty} a_n$, where $a_n = \ln(2n) - \ln(n+2)$. To analyze this series:

• Find the nth term: $a_n = \ln(2n) - \ln(n+2)$

• Partial sum formula: $S_n = \sum_{k=1}^n [\ln(2k) - \ln(k+2)]$

• Simplification using logarithm properties: $\ln(2k) - \ln(k+2) = \ln\left(\frac{2k}{k+2}\right)$

• Telescoping series: Many logarithmic series can be simplified by telescoping, where terms cancel out in the sum.

Example Calculation:

• Write out the first few terms to observe cancellation.

• Apply the limit as $n \to \infty$ to determine convergence.

Convergence Tests

Several tests are used to determine whether a series converges or diverges:

• Integral Test: If $a_n = f(n)$ and $f(x)$ is positive, continuous, and decreasing for $x \geq N$, then $\sum a_n$ converges if and only if $\int_N^{\infty} f(x) dx$ converges.

• Limit Comparison Test: Compare $a_n$ to a known series $b_n$:

  • If $\lim_{n \to \infty} \frac{a_n}{b_n} = L$ where $0 < L < \infty$, then both series converge or both diverge.

• Direct Comparison Test: If $0 \leq a_n \leq b_n$ and $\sum b_n$ converges, then $\sum a_n$ converges.

• Divergence Test: If $\lim_{n \to \infty} a_n \neq 0$, then $\sum a_n$ diverges.

Example: Geometric Series

A geometric series has the form $\sum_{n=0}^{\infty} ar^n$. It converges if $|r| < 1$ and diverges otherwise.

• Sum formula: $S = \frac{a}{1 - r}$ for $|r| < 1$

• Example: $\sum_{n=0}^{\infty} \left(-\frac{2}{3}\right)^n$ converges because $|r| = \frac{2}{3} < 1$

Example: p-Series

A p-series is of the form $\sum_{n=1}^{\infty} \frac{1}{n^p}$.

• Convergence: The series converges if $p > 1$ and diverges if $p \leq 1$.

• Example: $\sum_{n=1}^{\infty} \frac{1}{n^{3/2}}$ converges because $p = 3/2 > 1$.

Example: Series Involving Arctangent

Consider the series $\sum_{n=1}^{\infty} [\arctan(n+1) - \arctan(n)]$.

• Telescoping: The sum telescopes to $\arctan(n+1) - \arctan(1)$ as $n \to \infty$.

• Limit: $\lim_{n \to \infty} \arctan(n+1) = \frac{\pi}{2}$

• Partial sum: $S_n = \arctan(n+1) - \arctan(1)$

• Convergence: The series converges to $\frac{\pi}{2} - \arctan(1)$

Summary Table: Series Convergence Tests

Test Name	Condition	Converges?	Example
Integral Test	$f(x)$ positive, continuous, decreasing	If $\int f(x) dx$ converges	$\sum \frac{1}{n^{3/2}}$
Limit Comparison Test	$\lim_{n \to \infty} \frac{a_n}{b_n} = L$, $0 < L < \infty$	Both series behave the same	$\sum \frac{1}{n^p}$ vs. $\sum \frac{1}{n^{3/2}}$
Divergence Test	$\lim_{n \to \infty} a_n \neq 0$	Diverges	$\sum \frac{n}{n+1}$
Geometric Series	$|r| < 1$	Converges	$\sum \left(-\frac{2}{3}\right)^n$
p-Series	$p > 1$	Converges	$\sum \frac{1}{n^{3/2}}$

Worked Examples

• Find the nth partial sum: For $a_n = \ln(2n) - \ln(n+2)$, $S_n = \sum_{k=1}^n [\ln(2k) - \ln(k+2)]$

• Determine convergence: Use telescoping or a convergence test as appropriate.

• Geometric series: $\sum_{n=0}^{\infty} \left(-\frac{2}{3}\right)^n$ converges, sum is $\frac{1}{1 - (-2/3)} = \frac{1}{5/3} = \frac{3}{5}$

• p-series: $\sum_{n=1}^{\infty} \frac{1}{n^{3/2}}$ converges by the integral test.

• Arctangent series: $\sum_{n=1}^{\infty} [\arctan(n+1) - \arctan(n)]$ converges to $\frac{\pi}{2} - \arctan(1)$

Key Formulas

• $S_n = a_1 + a_2 + \cdots + a_n$

• Geometric series sum: $S = \frac{a}{1 - r}$

• p-series: $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges if $p > 1$

• Integral test: $\int_1^{\infty} f(x) dx$

Additional info: Some steps and justifications are inferred from standard calculus II techniques and the context of the provided questions.