Infinite Series Convergence

**MaxJasper** · August 16th 2012, 12:55 PM

Hi folks,

Please comment on how to proceed with finding whether the following series converge or not:
$1+\frac{1}{2}+\frac{1}{3}-\frac{1}{4}-\frac{1}{5}-\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}-\frac{1}{10}\text{--}\text{++}+\text{--}-$

Thanks a lot.

Max.

**Plato** · August 16th 2012, 02:38 PM

Originally Posted by MaxJasper

Please comment on how to proceed with finding whether the following series converge or not:
$1+\frac{1}{2}+\frac{1}{3}-\frac{1}{4}-\frac{1}{5}-\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}-\frac{1}{10}\text{--}\text{++}+\text{--}-$

This is only a comment or suggestion. Let
$\begin{align*}s_1&=\frac{1}{1}+\frac{1}{2}+\frac{1 }{3} \\ s_2&=\frac{1}{4}+\frac{1}{5}+\frac{1}{6} \\&\vdots \\ s_n&=\frac{1}{3n-2}+\frac{1}{3n-1}+\frac{1}{3n}\end{align*}$

Now your series is $\sum\limits_{n = 1}^\infty {{{\left( { - 1} \right)}^{n + 1}}{s_n}}$ .
Does the alternating series test apply?

**MaxJasper** · August 16th 2012, 02:57 PM

What a beauty! Thanks a lot bro...very simple and nice.

**GJA** · August 16th 2012, 03:05 PM

Had a response, but Plato's solution is very nice.

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