$$\sum_{n=1}^\infty(a_n-a_{n+k}).$$
I have been taught to call series of the above form with $k=1$ (and only with $k=1$) "telescopic", so when yesterday my friend who's studyin Economy and Commerce called such series "telescopic" regardless of $k$ I was a bit surprised. Is it common to call all of those simply "telescopic"? Or is there more specific terminology, something like (making this up) "$k$-telescopic" for the above form, and "telescopic" as an abbreviation of "1-telescopic"? Also, does the sequence $a_n$ have a special name, like e.g. generating sequence (again, made up by me)?
Personally, I call any series whose terms cancel a telescoping series. Of course, that's just my thought on the terminology.
As a related sidenote, we know that if $\lim\limits_{x\to\infty}f(x)=L$, then
$$\sum_{k=1}^nf(k)=nL+\sum_{k=1}^\infty f(k)-f(k+n)$$
So for any continuous function, say, $1/x$, we can have non-integer sums using telescoping series:
$$H_n=\sum_{k=1}^n\frac1k=\sum_{k=1}^\infty\frac1k-\frac1{k+n}$$
The last series being defined for non-integer values, such as $n=1/2$. Of course, such an extension is not unique, though it can be interesting and useful.