CK12 Calculus - Single Variable

s 3 =u 1 +u 2 +u 3

The first sum is the first term of the sequence.The secondsum is the sum of the first two terms.The third
term is the sum of the first threeterms.Thus,thenth finite sum,snis the sum of the firstntermsof the infinite
series:sn=u 1 + u 2 + u 3 + ... + un.

Sequenceof PartialSums
As you can see, the sumssn= u 1 + u 2 + u 3 +...+ un form a sequence.The sequenceis very importantfor
the studyof the relatedinfiniteseriesfor it tells a lot aboutthe infiniteseries.

PartialSumsFor an infiniteseries , then thpartialsum,snis the sum of the firstntermsof the

infiniteseries:. The sequence{sn} formedfrom thesesumsis calledthesequenceof partial

sums.

Example 2
Find the first five partialsumsof the infiniteseries1 + 0.1 + 0.01 + 0.001+ ....
Solution
s 1 =u 1 = 1

s 2 =u 1 + u 2 = 1 + 0.1 = 1.1

s 3 = 1 + 0.1 + 0.01 = 1.11

s 4 = 1 + 0.1 + 0.01 + 0.001+ 0.0001= 1.1111

s 5 = 1 + 0.1 + 0.01 + 0.001+ 0.0001= 1.1111

Convergenceand Divergence
Just as with sequences,we can talk aboutconvergenceand divergenceof infiniteseries.It turnsout that
the convergenceor divergenceof an infiniteseriesdependson the convergenceor divergenceof the se-
quenceof partialsums.

Convergence/Divergenceof SeriesLet be an infiniteseriesand let {sn} be the sequenceof

partialsumsfor the series.If {sn} has a finitelimitl, then the infiniteseriesconvergesand.

If {sn} doesnot havea finitelimit, then the infiniteseriesdiverges.The infiniteseriesdoesnot havea sum.

Example 3