CK12 Calculus - Single Variable

Example 11

Find the sum of.
Solution

By the RulesTheorem,. The series is a geometricserieswitha

= 5 and. The sum of is.

Then.
Addingor subtractinga finitenumberof termsfrom an infiniteseriesdoesnot affect convergenceor diver-
gence.

TheoremIf converges,then is also convergent.If con-

verges,then is also convergent.Likewise,if diverges,then

and are also divergent.

For a convergentseries,addingor removinga finitenumberof termswill not affect convergence,but it will
affect the sum.
Example 12

Find the sum of.
Solution

is a geometricserieswitha= 3 and. Its sum is

Then
Reindexing
Anotherpropertyof convergentseriesis that we canreindexa serieswithoutchangingits convergence.
This meanswe can start the indicesof the serieswith anothernumberotherthan 1. Keepthe termsin order