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 ∑ 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 an = f(n) where f is positive, continuous, and decreasing on [1,∞), then ∑ an and ∫_1^∞ 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: ∫1^∞ f(x) dx = lim{b→∞} ∫_1^b f(x) dx.

Conditions for the Integral Test

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

Partial sum (S_N)

The finite sum of the first N terms of a series: SN = ∑{n=1}^N a_n.

Series sum (S)

If a series converges, its sum S is the limit of partial sums: S = lim{N→∞} SN.

Remainder / error (R_N)

The difference between the true sum and the Nth partial sum: RN = S − SN.

p-series

A benchmark series of the form ∑_{n=1}^∞ 1/n^p, which converges if p > 1 and diverges if p ≤ 1.

Harmonic series

The series ∑_{n=1}^∞ 1/n, a p-series with p = 1 that diverges.

Geometric series

A series of the form ∑_{n=0}^∞ a r^n, which converges if |r| < 1 (and diverges if |r| ≥ 1).

Direct Comparison Test

If 0 ≤ an ≤ bn and ∑ bn converges, then ∑ an converges; if 0 ≤ bn ≤ an and ∑ bn diverges, then ∑ an diverges.

Limit Comparison Test

For positive terms, compute L = lim{n→∞} (an/bn). If 0 < L < ∞, then ∑ an and ∑ b_n either both converge or both diverge.

Inconclusive (Limit Comparison)

If L = 0 or L = ∞ 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 ∑ (-1)^{n-1} bn or ∑ (-1)^n bn with b_n ≥ 0.

Alternating Series Test (Leibniz Test)

An alternating series ∑ (-1)^{n-1} bn converges if bn is eventually decreasing and b_n → 0 as n → ∞.

nth-term divergence idea

If lim{n→∞} an ≠ 0 (or does not exist), then the series ∑ a_n must diverge.

Alternating Series Error Bound

If the Alternating Series Test applies, then the error after N terms satisfies |RN| ≤ 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 ∑ an converges absolutely if ∑ |an| converges.

Ratio Test

Compute L = lim{n→∞} |a{n+1}/a_n|. If L < 1 the series converges absolutely; if L > 1 (or L = ∞) it diverges; if L = 1 it is inconclusive.