CK12 Calculus - Single Variable

.

Then and.
OtherDivergentSeries(nth-Term Test)
Determiningconvergenceby usingthe limit of the sequenceof partialsumsis not alwaysfeasibleor practical.
For otherseries,it is moreusefulto applyteststo determineif an infiniteseriesconvergesor diverges.Here
are two theoremsthat help us determineconvergenceor divergence.

Theorem(Thenth-Term Test)If the infiniteseries converges,then TheoremIf

or if doesnot exist,then the infiniteseries diverges.

The first theoremtells us that if an infiniteseriesconverges,then the limit of the sequenceof termsis 0. The
converseis not true: If the limit of the sequenceof termsis 0, then the seriesconverges.So, we cannotuse
this theoremas a test of convergence.
The secondtheoremtells us that if limit of the sequenceof termsis not zero,then the infinitesseriesdiverges.
This givesus the first test of divergence:the n th-Term TestorDivergenceTest. Notethat if the test is
appliedand the limit of the sequenceof termsis 0, we cannotconcludeanythingand mustuse anothertest.
Example 9

Determineif convergesor diverges.
Solution
We can use thenth-Term Test to determineif the seriesdiverges.Thenwe do not haveto checkfor conver-
gence.

Because , the series diverges.
Example 10

Determineif convergesor diverges.
Solution