CK12 Calculus - Single Variable

The series is similarto. Usingthe ComparisonTest, for allk. The

series divergessinceit is ap-serieswith. By the ComparisonTest, also di-
verges.

The IntegralTest
Anotherusefultest for convergenceor divergenceof an infiniteserieswithoutnegativetermsis theIntegral
Test. It involvestakingthe integralof the functionrelatedto the formulain the series.It makessenseto use
this kind of test for certainseriesbecausethe integralis the limit of a certainseries.

Theorem(TheIntegralTest)Let be a serieswithoutnegativeterms.Iff(x) is a de-

creasing,continuous,non-negativefunctionforx≥ 1, then:1. convergesif and only if

converges.2. divergesif and only if diverges.

In the statementof the IntegralTest, we assumedthatukis a functionf ofk. We then changedthat function
fto be a continuousfunctionofxin orderto evaluatethe integraloff. If the integralis finite,then the infinite
seriesconverges.If the integralis infinite,the infiniteseriesdiverges.The convergenceor divergenceof
the infiniteseriesdependson the convergenceor divergenceof the correspondingintegral.

Example 7 Determineif convergesor diverges.
Solution
We can use the IntegralTest to determineconvergence.Write the integralform:

.

Next,evaluatethe integral.

Use the followingu-substitutionto evaluatethe integral:
u= 2x+ 1
du= 2dx