I'm studying Series and Diverge Test. But I'm having a problem understanding it. It says that, when the limit of it's partial sums is not equal to zero then it diverges. But then, there's also an example in my book where the limit of its partial sums approaches 1 and the writer is saying that it converges? I'm attaching the snapshot. Somebody help please, I've a quiz tomorrow.
You are confusing series with sequence, a common misunderstanding.
A sequence is a list (usually infinite) of numbers e.g.{n | n $\in \mathbb{N}$} ={1, 2, 3, 4, ...}.
A series is an infinite sum of numbers taken from a sequence e.g. $\sum_{n=1}^{\infty}n$ = 1 + 2 + 3 + 4 + ...
A partial sum is an approximation of the series, or a sum of a finite number of terms in the sum
e.g $S_4 = \sum_{k=1}^4n=1+2+3+4=10$
Then we say that a series converges to $L$ if the sequence of partial sums generated by $S_n=\sum_{k=1}^n a_k$ converges to $L$. That is, $\displaystyle\lim_{n \to \infty}S_n=L$.
The Divergence Tests tells us that if a series converges, then the limit of $a_n$ must be zero, otherwise the partial sums would continue to grow and $S_n$ would not converge to the finite number $L$. The equivalent statement (by contraposition) is: if the limit of a sequence is not zero, then the sum (series) diverges.
Here's an example:
$$\frac{1}{9} =0.1111\ldots=\displaystyle\sum_{n=1}^{\infty}\left(\frac{1}{10}\right)^n$$ $a_n= \{0.1,0.01,0.001, 0.0001,...\}, S_n=\{.1,.11,.111,.1111,...\}, L = \frac{1}{9}$
The divergence test has to do with $\displaystyle \lim_{n \to \infty}a_n \verb" not " \lim_{n \to \infty}S_n.$
In your example (a telescoping series), the series is given by $\sum_{n=1}^{\infty} a_n$. Where
$a_n = \frac{1}{n(n+1)} = \frac{1}{n}-\frac{1}{n+1}$. Note $\displaystyle \lim_{n \to \infty}a_n = 0$ and $\displaystyle \lim_{n \to \infty}S_n = \lim_{n \to \infty}1-\frac{1}{n+1} = 1 - 0 = L$
However, the first example I gave $\left(\sum_{n=1}^{\infty}n\right)$ diverges since $\displaystyle \lim_{n \to \infty} n = \infty \neq 0$
See more: http://en.wikipedia.org/wiki/Convergent_series