# Infinite Series Convergence

• Aug 16th 2012, 01:55 PM
MaxJasper
Infinite Series Convergence
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.
• Aug 16th 2012, 03:38 PM
Plato
Re: Infinite Series Convergence
Quote:

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?
• Aug 16th 2012, 03:57 PM
MaxJasper
Re: Infinite Series Convergence
What a beauty! Thanks a lot bro...very simple and nice.(Clapping)
• Aug 16th 2012, 04:05 PM
GJA
Re: Infinite Series Convergence
Had a response, but Plato's solution is very nice.