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Infinite Series Convergence Tests

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Alternating Series Estimation Theorem

Provides an estimate for the sum of an alternating series

Alternating Series: $\sum_{n=1}^{\infty} (-1)^n a_n$

Test: Alternating Series Test, Converges if $(a_n)$ is decreasing and $\lim_{n \to \infty} a_n = 0$

Comparison Test: $\sum a_n$ compared with $\sum b_n$ , where $0 \leq a_n \leq b_n$

If $\sum b_n$ converges, then $\sum a_n$ also converges; if $\sum a_n$ diverges, then $\sum b_n$ also diverges

Telescoping Series: $\sum (a_n - a_{n+1})$

Evaluates series by summing differences that cancel out successive terms, converges if the sequence of partial sums converges

P-Series: $\sum_{n=1}^{\infty} \frac{1}{n^p}$

Test: P-Series Test, Converges if $p > 1$

Geometric Series: $\sum_{n=0}^{\infty} ar^n$

Test: Geometric Series Test, Converges if $|r| < 1$

Conditional Convergence: $\sum a_n$

If $\sum a_n$ converges but $\sum |a_n|$ diverges, the series converges conditionally

Integral Test: $\sum a_n$ where $a_n = f(n), f$ is positive, continuous, and decreasing

Compares with the integral of $f$ , converges if the integral is finite, otherwise diverges

Ratio Test: $\sum a_n$ , using $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$

Converges if limit is less than 1, diverges if limit is greater than 1 or infinite, test is inconclusive if limit equals 1

Limit Comparison Test: $\sum a_n$ compared with $\sum b_n$ , using $\lim_{n \to \infty} \frac{a_n}{b_n}$

If the limit is positive and finite, both series converge or diverge together

Root Test: $\sum a_n$ , using $\lim_{n \to \infty} \sqrt[n]{|a_n|}$

Converges if limit is less than 1, diverges if limit is greater than 1 or infinite, test is inconclusive if limit equals 1

Cauchy Condensation Test: $\sum a_n$ for $a_n$ non-increasing and non-negative

$\sum a_n$ converges or diverges with $\sum 2^n a_{2^n}$

Harmonic Series: $\sum_{n=1}^{\infty} \frac{1}{n}$

No specific test, Diverges

Absolute Convergence Test: $\sum a_n$

If $\sum |a_n|$ converges, then $\sum a_n$ converges absolutely

Divergence Test: $\sum a_n$

If $\lim_{n \to \infty} a_n \neq 0$ , then $\sum a_n$ diverges

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