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Consider the series : $$2- \frac{3}{2} + \frac{2}{3}- \frac{1}{4} + \frac{2}{5} - \frac{3}{6} + \frac{2}{7} - \frac{1}{8} + \cdots$$

So I'm suppose to test convergence. Answer sheet says I can't apply the alternating test here. Okay, I can't but isn't alternating test just a corollary of the Dirichlet test? Why can't I apply the Dirichlet test here? Let's rewrite this series as $2-\sum a_n b_n$ and try computing the sum

Okay let $a_n=\frac{1}{n+1}$ and consider $b_n$ which goes like this :

$$-3 ,2,-1,2,-3,2,-1,-3.... $$

Now partial sums of $\sum b_n$ is bounded.

$a_n$ is monotonically decreasing. and $a_n \to 0$

So $\sum a_n b_n$ converges.

then $2- \sum a_n b_n$ converges.

What Am I doing wrong here?

  • 0
    Your answer sheet does not say one cannot apply the Dirichlet test, does it? So, *what is your question here?* – 2017-02-19
  • 0
    Why do you think Dirichlet doesn't apply? – 2017-02-19

1 Answers 1

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HINT :

$$\frac{2}{1} - \frac{3}{2} + \frac{2}{3} - \frac{1}{4} + \frac{2}{5} - \frac{3}{6} + \frac{2}{7} - \frac{1}{8} + \cdots$$

$$\frac{2}{1}- \frac{2+1}{2} + \frac{2}{3} - \frac{2-1}{4} + \frac{2}{5} - \frac{2+1}{6} + \frac{2}{7} - \frac{2-1}{8} + \cdots$$

$$\frac{2}{1} - \frac{2}{2} - \frac{1}{2} + \frac{2}{3} - \frac{2}{4} + \frac{1}{4} + \frac{2}{5} - \frac{2}{6} - \frac{1}{6} + \frac{2}{7} - \frac{2}{8} + \frac{1}{8} + \cdots$$

$$\frac{2}{1} - \frac{2}{2} + \frac{2}{3} - \frac{2}{4} + \frac{2}{5} - \frac{2}{6} + \frac{2}{7} - \frac{2}{8} +\cdots - \frac{1}{2} + \frac{1}{4} - \frac{1}{6} + \frac{1}{8}+\cdots $$