Is there any other definition of convergent series? As far as I know, convergence is only defined for sequences. And if we want to define convergence for series, we must first convert them into sequences. Clearly, the definition for convergent series with a sequence of partial sums is intuitive, but is this the only definition? Can the convergence of series be defined in some other way? If yes, why is this definition chosen? What properties do we get and what properties do we lose if we define convergence for series like this? Is there any other definition consistent with this one but more general? 
 A: It is by definition that:
$$\sum_{n=k}^\infty a_n=\lim_{N\to\infty}\sum_{n=k}^Na_n\tag1$$
There are no other definitions of convergence.  However, one may develop the idea of regularization.  For example, a series could be seen to be Abel summable.  In particular,
If $\displaystyle\sum_{n=k}^\infty a_n$ converges, then it is equal to $\displaystyle\lim_{x\to1^-}\sum_{n=k}^\infty a_nx^k$.  However, some divergent series happen to be Abel summable:
$$\lim_{x\to1^-}\sum_{n=0}^\infty(-1)^nx^n=\lim_{x\to1^-}\frac1{1+x}=\frac12$$
We also have Cesàro summability.  If $\displaystyle\sum_{n=1}^\infty a_n$ converges, then it is equal to $\displaystyle\lim_{N\to\infty}\frac1N\sum_{k=1}^N\sum_{n=1}^ka_n$.  Again, we find some divergent series to be Cesàro summable:
$$\lim_{N\to\infty}\frac1N\sum_{k=1}^N\sum_{n=1}^k(-1)^{n+1}=\lim_{N\to\infty}\frac1N\sum_{k=1}^N\delta_{1,~k\pmod2}=\frac12$$
Other interesting things include analytic continuation, Ramanujan summable, and Borel summable.

Warning!
While a series may be Abel summable, Cesàro summable, etc., it does not mean it converges.  Convergence is solely defined by $(1)$, unless you are in the realm of p-adic numbers.
