Consider a general term  a_n  representing the amount of something that accumulates during the  n ^th  interval of time, after some starting point. The sequence  {a_n}  describes the amounts that accumulate during the first interval, the second interval, the third interval, etc.

The total accumulation over the first  N  intervals is then given by:

S_N = a₁ + a₂ + a₃ + ... + a_N

We call this the  N ^th  partial sum of the sequence. The usual shorthand is:

where the Greek letter sigma, S , stands for "sum", and the numbers below and above specify a consecutive range of inputs.

If we imagine the accumulation continuing indefinitely, we write:

This is called an infinite series. A series is just a sum with an unlimited number of summands. For short, we often write this as  S a_n .

Notice that each sequence  {a_n}  determines a series  S a_n , and each series  S a_n  determines a sequence  {S_N} . The two ideas are obviously very closely related.

We say that a series converges when its sequence of partial sums converges. If the partial sums "settle down" to some value  S , then it makes sense to talk about "the sum"  S = S a_n  of the series, even though the number of summands is infinite.

Consider, for example, the two series:

Plots of the first 100 partial sums look like this:

While both sums continue to grow as more terms are added, it appears that in the first case the partial sums may be "settling down" to a sum  S  somewhere between  1.6  and  1.7 . Indeed, the first series converges (to  p²/6) and the second one diverges.

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