Where To Start The Index Of A Sum

When we look at infinite sums, they almost always start the index at 0 or 1. For example, a geometric series looks like

    \[ \sum_{n=1}^{\infty} ar^{n-1} \]

and the number e is equal to

    \[ \sum_{i=0}^{\infty} \frac{1}{i!}. \]

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

    \[ \sum_{k=117}^{\infty} ar^{k-117}, \]

and instead of the second I could have written

    \[ \sum_{i=-3}^{\infty} \frac{1}{(i+3)!}. \]

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:

    \[ \sum_{i=1}^{\infty} a_i = \sum_{m=1}^{\infty} a_m. \]

These are just cosmetic issues, and don’t affect the series.

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