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1.33 Series

A sequence is a list, whereas a series is a sum.

Summation notation. Let $(a_{n})$ be a sequence, indexed by $n$ . We can write

\sum_{n=m}^{k}a_{n}=a_{m}+a_{m+1}+\dots+a_{k}

for the sum of these terms, beginning with the index $m$ and stopping with index $k$ ; by convention, we take $m\leq k$ .

$\bullet$ $n$ runs;

$\bullet$ start at $m$ ;

$\bullet$ stop at $k$ .

Note that $n$ is the running index that we sum over, and that the final sum does not involve $n$ , but clearly the right-hand side depends upon both $m$ and $k$ . We must use a different symbol for the running index $n$ than for the terms $m$ and $k$ .