Infinite Series: every key term you need (+ practice quiz)

More AP Calculus BC guides

• Limits & Continuity
• Derivatives
• Applications of Derivatives
• Integration
• Applications of Integrals

Sequences

Ordered list of numbers a₁, a₂, a₃, ... Converges if lim(n→∞) aₙ = L. Diverges otherwise.

Series Definition

Sum of sequence terms: S = Σ aₙ. Partial sum: S_N = a₁ + a₂ + ... + aₙ.

Geometric Series

Σ ar^(n-1) = a/(1-r) if |r|<1. Convergent if |r|<1, divergent otherwise.

Arithmetic Series

Σ (first term + common diff·n). Diverges unless common diff = 0.

Divergence Test

If lim aₙ ≠ 0, series diverges. If lim aₙ = 0, test inconclusive (series may converge/diverge).

Integral Test

If ∫f(x)dx converges, Σf(n) converges. If integral diverges, series diverges. f must be positive, decreasing.

p-Series

Σ 1/n^p converges if p>1, diverges if p≤1. Example: Σ 1/n² converges, Σ 1/n diverges.

Comparison Test

If 0 < aₙ ≤ bₙ and Σbₙ converges, then Σaₙ converges. If 0 < cₙ ≤ aₙ and Σcₙ diverges, then Σaₙ diverges.

Alternating Series

Σ (-1)^n aₙ. Converges if aₙ decreasing and lim aₙ = 0 (alternating series test).

Absolute vs Conditional Convergence

Absolute: Σ|aₙ| converges. Conditional: Σaₙ converges but Σ|aₙ| doesn't.

Power Series

Σ aₙ(x-c)^n. Converges for |x-c| < R (radius of convergence). Found using ratio test.

Unit 7 Summary

Series: sum of sequence. Tests for convergence: divergence, integral, p-series, comparison, alternating. Power series in terms of x.

Sequence vs Series

A sequence is a list of terms; a series is the sum of a sequence. A series converges when its sequence of partial sums converges.

nth Term Test

If lim a_n ≠ 0, the series diverges. If the limit is 0, the test is inconclusive — the harmonic series proves this.

Geometric Series

Σ ar^n converges to a/(1-r) exactly when |r| < 1, where a is the first term actually present in the sum.

p-Series

Σ 1/n^p converges when p > 1 and diverges when p ≤ 1. The p=1 harmonic case diverges despite terms tending to zero.

Integral Test

For positive decreasing continuous f with f(n)=a_n, the series and ∫(1 to ∞)f dx converge or diverge together. It supplies the p-series result.

Direct Comparison Test

If 0 ≤ a_n ≤ b_n and Σb_n converges then Σa_n converges; if Σa_n diverges then so does Σb_n. The inequality must point the right way.

Limit Comparison Test

If lim a_n/b_n is a finite positive number, both series share their fate. Choose b_n from the dominant terms of a_n.

Alternating Series Test

An alternating series converges if the term magnitudes decrease monotonically to zero. Both conditions are required.

Alternating Series Error Bound

The truncation error is at most the magnitude of the first omitted term, and it has that term's sign.

Absolute vs Conditional

If Σ|a_n| converges the series converges absolutely. If Σa_n converges but Σ|a_n| does not, the convergence is conditional, as for the alternating harmonic series.

Ratio Test

Compute L = lim |a_(n+1)/a_n|. L < 1 converges absolutely, L > 1 diverges, L = 1 gives no information.

Root Test

L = lim |a_n|^(1/n) with the same conclusions as the ratio test; it is preferable when the nth term is an nth power.

Radius and Interval of Convergence

The ratio test on a power series gives the radius R; the endpoints must then be tested individually by other tests.

Endpoint Behavior

A power series may converge absolutely at one endpoint, conditionally at the other, or at neither — each must be checked by substitution.

Term-by-Term Operations

Inside the interval of convergence a power series may be differentiated and integrated term by term, with the same radius but possibly different endpoint behavior.

Telescoping Series

Partial fractions can collapse a sum so that most terms cancel; the limit of the surviving partial sum is the value of the series.

Rearrangement Danger

Conditionally convergent series can be rearranged to sum to any value, so term order matters; absolutely convergent series are safe to rearrange.

Choosing a Test

Check the nth term first, then look for geometric or p-form, then ratio for factorials and powers, comparison for rational expressions, and alternating last.

Series of Positive Terms

For positive terms the partial sums increase, so convergence is equivalent to the partial sums being bounded above.

Factorial Growth

n! eventually dominates any exponential c^n, which is why Σ c^n/n! converges for every c and sums to e^c.