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Calculus Sequences and Series:
Problems and Solutions
When we look at infinite sums, they almost always start the index at 0 or 1. For example, a geometric series looks like
![\[ \sum_{n=1}^{\infty} ar^{n-1} \]](http://thecalculusblog.com/wp-content/ql-cache/quicklatex.com-a936265cc634381399b5e0093be0460c_l3.png "Rendered by QuickLaTeX.com")
and the number 
is equal to
![\[ \sum_{i=0}^{\infty} \frac{1}{i!}. \]](http://thecalculusblog.com/wp-content/ql-cache/quicklatex.com-11aaa0910a1378d7143ebf947e74ddc6_l3.png "Rendered by QuickLaTeX.com")
So do sums have to start with 0 or 1? No! They can start wherever we like. It just happens that 0 and 1 are often convenient values. Instead of the first sum above, I could have written
![\[ \sum_{k=117}^{\infty} ar^{k-117}, \]](http://thecalculusblog.com/wp-content/ql-cache/quicklatex.com-1347693a7b713f956ba10d9209917190_l3.png "Rendered by QuickLaTeX.com")
and instead of the second I could have written
![\[ \sum_{i=-3}^{\infty} \frac{1}{(i+3)!}. \]](http://thecalculusblog.com/wp-content/ql-cache/quicklatex.com-47bbccc15b71be885d23b479d93728b5_l3.png "Rendered by QuickLaTeX.com")
The sums are the same. The reason we usually don’t start with 117 or -3 is that these are goofy values. The numbers 0 and 1 are just prettier 
Note also that the name of the indexing variable doesn’t matter:
![\[ \sum_{i=1}^{\infty} a_i = \sum_{m=1}^{\infty} a_m. \]](http://thecalculusblog.com/wp-content/ql-cache/quicklatex.com-1186167b2bb1095ebf09f74d2a2baf88_l3.png "Rendered by QuickLaTeX.com")
These are just cosmetic issues, and don’t affect the series.