Series: A Series (Or Infinite Series) is obtained from a sequence by adding the terms of the sequence. Another sequence associated with the series is the sequence of partial sums. A series converges if its sequence of partial sums converges. The sum of the series is the limit of the sequence of partial sums.
For example, consider the geometric series defined by the sequence $$a_n= \frac{1}{r^n}, \quad n=0,1,2, \ldots$$ Then the $n$-th partial sum $S_n$ is given by
$\displaystyle S_n = \sum_{k=0}^n \frac{1}{r^k} =$ and, for $|r| > 1$,
$\displaystyle \sum_{k=0}^\infty \frac{1}{r^k} = \lim_{n\longrightarrow\infty} S_n =$ .
Thus
$\sum_{k=0}^\infty \left(\frac{2}{\pi}\right)^k =$ and
$\sum_{k=2}^\infty \left(\frac{2}{\pi}\right)^k =$ .
Another situation in which we we can actually compute the partial sums occurs if those sums are collapsing. It may not be obvious that that is the case, but look for it in this example:
$\sum_{k=0}^n \left(\frac{1}{k^2+9k+20}\right)$ = and thus
$\sum_{k=0}^\infty \left(\frac{1}{k^2+9k+20}\right)$ = .