CK12 Calculus - Single Variable

Rewrite as to get. The valueofpis. This is less than 1, whichtells
us that the seriesdiverges.

ComparisonTest
Now that we havestudiedserieswithoutnegativeterms,we can applyconvergencetestsmadefor such
series.The first test we will consideris theComparisonTest.The nameof the test tells us that we will
compareseriesto determineconvergenceor divergence.

Theorem(The ComparisonTest)Let and be serieswithoutnegativeterms.Supposethat

u 1 ≤v 1 ,u 2 ≤v 2 , ...,ui≤vi, .... 1. If converges,then converges.2. If diverges,

then diverges.

In orderto use this test, we mustcheckthat for eachindexk, everyukis less than or equaltovk. This is

the comparisonpart of the test. If the serieswith the greaterterms, , converges,than the serieswith

the lesserterms, , converges.If the lesserseriesdiverges,then the greaterserieswill diverge.You
can only use the test in the ordersgivenfor convergenceor divergence.You cannotuse this test to say, for
example,that if the greaterseriesdiverges,than the lesserseriesalso diverges.

Example 5 Determinewhether convergesor diverges.
Solution

lookssimilarto , so we will try to applythe ComparisonTest. Beginby comparing

eachterm.For eachk, is less than or equalto , so. Since

is a convergentp-series,then,by the ComparisonTest, also converges.

Example 6 Determinewhether convergesor diverges.
Solution