CK12 Calculus - Single Variable

• Computethe sum of a convergentgeometricseries


• Determineconvergenceor divergenceof seriesusingthe nth-Term Test

InfiniteSeries(series,sequenceof partialsums,convergence,divergence)
Series
Anothertopicthat involvesan infinitenumberof termsis the topicofinfiniteseries. We can represent
certainfunctionsand numberswith an infiniteseries.For example,any real numberthat can be writtenas
a non-terminatingdecimalcan be representedas an infiniteseries.
Example 1

The rationalnumber can be writtenas 0.44444....We can expandthe decimalnotationas an infiniteseries:

On the otherhand,the number can be writtenas 0.25.If we expandthe decimalnotation,we get a finite
series:

Do you see the differencebetweenan infiniteseriesand a finiteseries?Let’s definewhatwe meanby an
infiniteseries.

InfiniteSeriesAn infiniteseriesis the sum of an infinitenumberof terms,u 1 , u 2 , u 3 , u 4 ,..., usuallywritten

as.u 1 + u 2 + u 3 + u 4 +....A shorthandnotationfor an infiniteseriesis to use sigmanotation: , which

can be read as “the sum of the termsuk’s forkequalto 1 to infinity.”

We can makefinitesumsfrom the termsof the infiniteseries:
s 1 =u 1

s 2 =u 1 +u 2