Section 11.5 – Testing for Convergence at Endpoints

Common Series to be used…. Harmonic Series - DIVERGES p-Series Converges if p > 1 Diverges if p < 1 Comparison Test for ConvergenceComparison Test for Divergence

acts like converges by the comparison test to the p-series with p = 2. Makes terms smaller

diverges by the nth term test for divergence. acts like diverges by the comparison test to the p-series with p = 1.

ABSOLUTE CONVERGENCE

Alternating Series Test converges if both of the following conditions are satisfied: If a series converges but the series of absolute values diverges, We say the series converges conditionally. CONDITIONAL CONVERGENCE – (AST)

Determine whether the series is converges conditionally, converges absolutely, or diverges. The series converges conditionally

Determine whether the series is converges conditionally, converges absolutely, or diverges. Determine whether the series is converges conditionally, converges absolutely, or diverges. The series diverges by the nth term test for divergence

Find the interval of convergence for If x = -1, converges, alternating series test If x = 1, diverges, harmonic series Interval of convergence [-1, 1) ? ?

Find the interval of convergence for If x = -7, diverges, nth term test If x = 1, diverges, nth term test Interval of convergence (-7, 1) ? ?

Find the interval of convergence for Interval of convergence

Find the interval of convergence for If x = 2,diverges, harmonic series If x = 4, converges, alternating series test Interval of convergence (2, 4] ??

Find an upper bound for the error if the sum of the first four terms is used as an approximation to the sum of the series.

Find the smallest value of n for which the nth partial sum approximates the sum of the series within