Help understanding the sequence of partial sums of a series
Matthew Martinez 
I just completed calc 2 and am compiling and rewriting my notes for future reference. One thing I found online (Wikipedia etc) but not really covered in my texts is the concept that a series contains a sequence of partial sums. How is it that a sequence of partial sums containing the cumulative set of every nth partial sum converges to the sum of the series? What I've read makes it sound like the sum of the sequence of nth partial sums is what the whole series converges to, but that seems wrong.
Trivial ex: a finite series 1+2+3+4 has sequence of nth partial sums {1,3,6,10} and the sum of that sequence is 20. Yet the sum of the actual series is 10.
So is it that the series sums to the last term in the sequence of nth partial sums? Or is there something else going on? Thanks.
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$\begingroup$The solution to the sum is not the sum of each of the partial sums, but rather the number to which the sequence of partial sums converges. This has little application for finite series, but for infinite series, this is how the infinite sum is defined:
Definition: The infinite sum is defined as the limit of the sequence of partial sums, if it converges. $$\sum_{n=1}^\infty a_n=\lim_{N\to\infty}\sum_{n=1}^N a_n.$$
For example, if the sum is $$\sum_{n=1}^\infty 1/2^n,$$ the partial sums are $$1/2,3/4,7/8,15/16,...$$ which unsuprisingly converge to $1$.
$\endgroup$ 1 $\begingroup$You absolutely don't want to start summing the partial sums, as your "trivial example" makes clear.
Partial sums are used to define the value of an infinite series. Consider a sequence $(x_i)$, and suppose I want to make sense of $\Sigma_i x_i$. Then I define another sequence, the sequence of partial sums, to be $s_n = \Sigma_i^n x_i$. If this sequence converges then I say that the series $\Sigma_i x_i$ converges, and I define its value to be $\lim_n s_n$. Notice there's no mention of adding the $s_n$'s. I hope that helps.
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