AP Calculus BC Unit 10: Convergence Tests for Infinite Series

Convergence test

A method for deciding whether an infinite series has a finite sum (converges) or does not settle to a finite value (diverges).

Infinite series

An expression of the form $\sum a_n$ representing the sum of infinitely many terms, interpreted as the limit of its partial sums.

Converges

A series converges if its sequence of partial sums approaches a finite limit.

Diverges

A series diverges if its partial sums do not approach a finite limit (may grow without bound or oscillate without settling).

Integral Test

A test stating that if $a_n = f(n)$ where f is positive, continuous, and decreasing on $[1,\infty)$, then $\sum a_n$ and $\int_1^\infty f(x)\,dx$ either both converge or both diverge.

Improper integral (to infinity)

An integral with an infinite limit of integration, evaluated using a limit: $\int_1^\infty f(x)\,dx = \lim_{b\to\infty} \int_1^b f(x)\,dx$.

Conditions for the Integral Test

The function f(x) must be positive, continuous, and decreasing for x $\ge$ 1, and satisfy $a_n = f(n)$.

Partial sum (S_N)

The finite sum of the first N terms of a series: $S_N = \sum_{n=1}^N a_n$.

Series sum (S)

If a series converges, its sum S is the limit of partial sums: $S = \lim_{N\to\infty} S_N$.

Remainder / error (R_N)

The difference between the true sum and the Nth partial sum: $R_N = S - S_N$.

p-series

A benchmark series of the form $\sum_{n=1}^\infty 1/n^p$, which converges if p > 1 and diverges if $p \le 1$.

Harmonic series

The series $\sum_{n=1}^\infty 1/n$, a p-series with p = 1 that diverges.

Geometric series

A series of the form $\sum_{n=0}^\infty a r^n$, which converges if |r| < 1 (and diverges if $|r| \ge 1$).

Direct Comparison Test

If $0 \le a_n \le b_n$ and $\sum b_n$ converges, then $\sum a_n$ converges; if $0 \le b_n \le a_n$ and $\sum b_n$ diverges, then $\sum a_n$ diverges.

Limit Comparison Test

For positive terms, compute $L = \lim_{n\to\infty} (a_n/b_n)$. If $0 < L < \infty$, then $\sum a_n$ and $\sum b_n$ either both converge or both diverge.

Inconclusive (Limit Comparison)

If L = 0 or L = $\infty$ in the Limit Comparison Test, the test does not determine convergence/divergence.

Dominant term (for comparisons)

The leading/most significant part of an expression for large n (often highest power of n) used to choose a comparison series $b_n$.

Alternating series

A series whose terms change sign, often written as $\sum (-1)^{n-1} b_n$ or $\sum (-1)^n b_n$ with $b_n \ge 0$.

Alternating Series Test (Leibniz Test)

An alternating series $\sum (-1)^{n-1} b_n$ converges if $b_n$ is eventually decreasing and $b_n \to 0$ as $n \to \infty$.

nth-term divergence idea

If $\lim_{n\to\infty} a_n \ne 0$ (or does not exist), then the series $\sum a_n$ must diverge.

Alternating Series Error Bound

If the Alternating Series Test applies, then the error after N terms satisfies $|R_N| \le b_{N+1}$ (the next term’s magnitude).

Tolerance (alternating approximation)

A desired maximum error; for alternating series you choose N so that $b_{N+1}$ < tolerance.

Conditional convergence

A series that converges, but its series of absolute values diverges (often occurs with alternating series).

Absolute convergence

A series $\sum a_n$ converges absolutely if $\sum |a_n|$ converges.

Ratio Test

Compute $L = \lim_{n\to\infty} |a_{n+1}/a_n|$. If L < 1 the series converges absolutely; if L > 1 (or $L = \infty$) it diverges; if L = 1 it is inconclusive.