AP Calculus BC Review

pls get a 5

nth - Term Test

Series will diverge when the limit of the terms does not approach zero as n approaches infinity.

A geometric series will converge when…

0 < absolute value of r < 1

A geometric series will diverge when…

the absolute value of r is greater than or equal to 1.

How do you find the sum of a geometric series?

S = a / (1 - r)

Alternating Series Test

An alternating series converges if the absolute value of |a_n+1| is less than |a_n| term AND the limit of a_n equals 0 as n approaches infinity.

What criteria must be met to use the integral test on a series?

The integral test can be applied if the function is continuous, positive, and decreasing on the interval being considered.

When will a p-series converge?

A p-series converges if the p-value is greater than 1.

When will a p-series diverge?

A p-series diverges if the p-value is less than or equal to 1.

How do you use the direct comparison test?

The direct comparison test involves comparing a given series to a known benchmark series to determine convergence or divergence.

When using the limit comparison test and we are given a series a_n, how do we find b_n?

Choose a positive series—using the direct comparison test—that resembles a_n

When will a series converge using the Limit Comparison Test?

If the limit as n approaches infinity of the ratio of a_n to b_n is positive and finite AND b_n converges.

When will a series diverge using the Limit Comparison Test?

If the limit as n approaches infinity of the ratio of a_n to b_n is positive and finite AND b_n diverges.

In regards to Absolute Convergence, the series a_n will absolutely converge if…

the series |a_n| converges.

How do we test for conditional convergence?

The series a_n converges but the series |a_n| diverges.

In regards to the Ratio Test, when will a series converge?

The series converges if the limit of the ratio of consecutive terms, |a_n+1/a_n|, is less than 1 as n approaches infinity.

In regards to the Ratio Test, when will a series diverge?

The series diverges if the limit of the ratio of consecutive terms, |a_n+1/a_n|, is greater than 1 or undefined as n approaches infinity.

When will the Ratio Test be inconclusive?

The Ratio Test is inconclusive if the limit of the ratio of consecutive terms, |a_n+1/a_n|, equals 1 as n approaches infinity.

When will a Power Series converge for only at c?

A power series will only converge at c IF lim_n→∞ |a_n+1/a_n| = infinity.

When will a power series converge absolutely for all x?

A power series will only converge at c IF lim_n→∞ |a_n+1/a_n| = 0.

What is the Error Bound for an Alternating Series?

|S - S_n| < |a_n+1|