Infinite Series

Series Basics

$\sum_{n=1}^\infty a_n = a_1 + a_2 + a_3 + \cdots$

A series converges if its partial sums approach a finite limit.

Divergence Test

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

(If $\lim a_n = 0$, the test is inconclusive.)

Geometric Series

$\sum_{n=0}^\infty ar^n = \frac{a}{1-r} \quad \text{if } |r| < 1$ Diverges if $|r| \geq 1$.

p-Series

$\sum_{n=1}^\infty \frac{1}{n^p}$ Converges if $p > 1$; diverges if $p \leq 1$.

Integral Test

If $f(n) = a_n$ is positive, continuous, and decreasing: $\sum a_n \text{ and } \int_1^\infty f(x)\,dx$ converge or diverge together.

Comparison Tests

• Direct Comparison: Compare to a known series.
• Limit Comparison: $\lim \frac{a_n}{b_n} = L > 0$ → both converge or both diverge.

Ratio Test

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

• $L < 1$: converges absolutely.
• $L > 1$: diverges.
• $L = 1$: inconclusive.

Root Test

$L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$ Same conclusions as ratio test.

Alternating Series Test

$\sum (-1)^n b_n$ converges if $b_n > 0$, $b_n$ is decreasing, and $\lim b_n = 0$.

Error bound: $|R_n| \leq b_{n+1}$.

Absolute vs. Conditional Convergence

• Absolute convergence: $\sum |a_n|$ converges.
• Conditional convergence: $\sum a_n$ converges but $\sum |a_n|$ diverges.

Problem: Does $\sum_{n=1}^\infty \frac{n}{2^n}$ converge?

Solution: Ratio test: $L = \lim \frac{(n+1)/2^{n+1}}{n/2^n} = \lim \frac{n+1}{2n} = \frac{1}{2} < 1$

The series converges.

• Start with the divergence test; if $\lim a_n \neq 0$, it diverges.
• Geometric: $|r|<1$ converges. p-series: $p>1$ converges.
• Ratio test is especially useful for factorials and exponentials.
• Alternating series: check decreasing terms with limit zero; error ≤ first omitted term.