Explanation on Series (Divergence Test) [ Solved]

Question

So, I'm a bit baffle on this concept for the convergence of series.

Test for Divergence: http://imgur.com/XZk547T (2/5)≠0 So, it's divergent.

Now, let's take a look at this example: http://imgur.com/EWYc8Dk

For this example, it converges to -(8/7). What? Did I miss something? I thought it had to be zero in order to converge?

Answer 1

Every series has two sequences associated with it.

If our series is $S = \displaystyle\sum_{n=1}^{+\infty} a_n$, then the two sequences are:

• $a_1, a_2, a_3, \dots$. This is the sequence of the terms of the series.
• $S_1, S_2, S_3, \dots$. This is the sequence of partial sums of the series. We define these as $S_k = \displaystyle\sum_{n=1}^k a_n$. For example, $S_3 = a_1 + a_2 + a_3$.

I thought it had to be zero in order to converge

The limit of the sequence must be zero. In other words, $\displaystyle\lim_{n\to+\infty} a_n = 0$ must be true.

The series itself could potentially take on any value. In other words, $\displaystyle \lim_{n\to+\infty} S_n$ is not necessarily zero since a series could converge to a nonzero number.

For example, $\displaystyle \sum_{n=1}^{+\infty} \sin n$ diverges, because $\displaystyle \lim_{n\to+\infty} \sin n$ does not exist (and therefore is not zero).

Another example is $\displaystyle \sum_{n=1}^{+\infty} \frac1{n^2} = \frac{\pi^2}6$.

WARNING: The limit of the sequence being zero is not a sufficient condition for convergence. For example, $\displaystyle \sum_{n=1}^{+\infty}\frac1n$ diverges even though $\displaystyle \lim_{n\to+\infty}\frac1n = 0$.

Answer 2

By the definition of convergence of a series:

$\sum_{n=0}^\infty a_n$ converges if $L=\lim_{N\to\infty}s_N=\lim_{N\to\infty}\sum_{n=0}^N a_n$ exists.

It then follows that if $\lim_{N\to\infty}s_N$ exists, then

$\lim_{N\to\infty}a_N=\lim_{N\to\infty}s_N-s_{N-1}=L-L=0$

Thus, for a series to converge, each individual term must approach zero.

However, the converse is not true. Just because each term goes to zero does not mean the series will converge.