# "Testing Convergence for Endpoints: Harmonic and p-Series"  Learn how to test for convergence of harmonic and p series (p>1 or p<1) at endpoints using the comparison test for convergence and divergence, and how to make terms smaller.

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Slide2Common Series to be used….Harmonic Series - DIVERGES p-Series Converges if p > 1 Diverges if p  <  1 Comparison Test for Convergence Comparison Test for Divergence

Slide3acts likeconverges by the comparison test to the p-series with p = 2. Makes terms smaller Makes terms smaller

Slide4diverges by the nth term testfor divergence. acts like diverges by the comparison test to the p-series with p = 1.

Slide5ABSOLUTE CONVERGENCE

Slide8Alternating Series Testconverges 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)

Slide9determine whether the series is converges conditionally,converges absolutely, or diverges. The series converges conditionally

Slide10determine 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

Slide13Find the interval of convergence forIf x = -1, converges , alternating series test If x = 1, diverges , harmonic series Interval of convergence  [ -1, 1 ) ? ?

Slide14Find the interval of convergence forIf x = -7, diverges , nth term test If x = 1, diverges , nth term test Interval of convergence  ( -7, 1 ) ? ?

Slide15Find the interval of convergence forInterval of convergence

Slide16Find the interval of convergence forIf x = 2, diverges , harmonic series If x = 4, converges , alternating series test Interval of convergence  ( 2, 4 ] ? ?

Slide17find an upper bound for the error if the sum of the first fourterms is used as an approximation to the sum of the series.

Slide18find the smallest value of n for which the nth partial sumapproximates the sum of the series within 0.005.