The originalproblem, , doesnot havethe leading
term of 3. This doesnot affect the convergencebut will affect the sum of the series.We needto subtract 3
from the sum of the series to get the sum of
.The sum of the seriesis:.
OtherConvergentSeries
Thereare otherinfiniteseriesthat will converge.
Example 8
Determineif convergesor diverges.If it converges,find the sum.
Solution
Thenth partialsumsn is:
We can simplifysn further. Noticethat the first parentheseshas whilethe secondparentheseshas
. Thesewill add up to 0 and cancelout. Likewise,the and
will cancelout. Continuein this way to canceloppositeterms.This sum is atelescopingsum, whichis a
sum of termsthat canceleachotherout so that the sum will fold neatlylike a foldingtelescope.Thus,we
can writethe partialsum as