AP Calc Unit 10 Study Guide

What is a monotonic sequence?

A sequence that constantly increases or constantly decreases

What is a convergent series?

A series with a partial sum that converges to a value as n approaches infinity.

What is a geometric sequence?

A sequence where the same number is multiplied to each term to get the next in the sequence.

When and where does a geometric series converge?

When the absolute value of r is less than 1, the series converges to ar^k÷(1-r) where ar is the first term.

What is the nth term test for divergence?

If the limit of the nth term of a series as n approaches infinity doesn’t equal to 0, the series diverges.

What is the integral test for convergence?

If f is a positive, continuous, and decreasing function for x ≥ k and a_n = f(x), then the sum of all terms in the series starting from n = k and the integral from k to infinity of f(x) both converge or both diverge.

When does a p series converge or diverge?

Converges if p > 1

Diverges if 0 < p ≤ 1

Do harmonic series converge or diverge?

Always diverge

What is the comparison test for convergence?

If 0 < a_n ≤ b_n for all n, then if the partial sum of b converges, the partial sum of a must converge. If the partial sum of a diverges, the partial sum of b must diverge.

What is the Limit Comparison Test?

If both a_n and b_n are greater than 0 and the limit of a_n divided by b_n as n approaches infinity is finite and positive, then the partial sums of a and b either both converge or both diverge.

What is the Alternating Series Test?

If a_n > 0, the limit as n approaches infinity of a_n = 0, and |a_n+1| ≤ |a_n| for all n, then the alternating series must converge. Otherwise, it must diverge.

What is the Ratio Test for Convergence?

If an infinite series a_n has positive terms, then if the limit of a divided by