I would suggest you read the book "An introduction to the Theory of Infinite Series" by Thomas Bromwich.
P.S.: There is a tale that goes something like Ramanujan was advised by British Mathematicians to read this book when he wanted to prove that the sum of a series of positive terms is negative ( I neither remember the series, nor Ramanujan's "sum" but all I could remember is it was divergent").
An online version, downloadable in many different formats, is here on the archive.com .
Read Comment below by Sivaram where he points out both the series and the sum!
The operation of addition is a binary operation: it is an operation defined on pairs of real (or complex) numbers. When we write something like $a+b+c$, apparently adding three numbers, we’re really doing repeated addition of two numbers, either $(a+b)+c$ or $a+(b+c)$ (assuming that we don’t change the order of the terms); one of the basic properties of this operation is that it doesn’t actually matter in which order we do these repeated binary additions, because they all yield the same result.
It’s easy enough to understand what it means to do two successive additions to get $a+b+c$, or $200$ to get $a_0+a_1+\ldots+a_{200}$; it’s not so clear what it means to do infinitely many of them to get $\sum_{k\ge 0}a_k$. The best way that’s been found to give this concept meaning is to define this sum to be the limit of the finite partial sums:
$$\sum_{k\ge 0}a_k=\lim_{n\to\infty}\sum_{k=0}^na_k\tag{1}$$
provided that the limit exists. For each $n$ the summation inside the limit on the righthand side of $(1)$ is an ordinary finite sum, the result of performing $n$ ordinary binary additions. This is always a meaningful object. The limit may or may not exist; when it does, it’s a meaningful object, too, but it’s the outcome of a new kind of operation. It is not the result of an infinite string of binary additions; we don’t even try to define such a thing directly. Rather, we look at finite sums, which we can define directly from the ordinary binary operation of addition, and then take their limit. In doing this we combine an algebraic notion, addition, with an analytic notion, that of taking a limit.
Finite sums like $a_0+\ldots+a_{200}$ all behave the same way: they always exist, and we can shuffle the terms as we like without changing the sum. Infinite series do not behave the same way: $\sum_{n\ge 0}a_n$ does not always exist, and shuffling the order of the terms can in some cases change the value. This really is a new operation, with different properties.
Best Answer
Write out a few terms of the series. You should see a pattern! But first consider the finite series:
$$\sum\limits_{n=1}^{m}\left(\frac{1}{n}-\frac{1}{n+1}\right) = 1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + \cdots + \frac{1}{m-1} - \frac{1}{m} + \frac{1}{m} - \frac{1}{m+1}.$$ This sum is telescoping, since it collapses like a telescope.
Everything is left except for the first and last term. Now what's the limit as $m\to \infty$?