alternating series: x - y/2 + x/3 - y/4 + x/5 - y/6 + ..., x,y > 0

[gathe335]

Given the following:

. . .\(\displaystyle x\, -\, \dfrac{y}{2}\, +\, \dfrac{x}{3}\, -\, \dfrac{y}{4}\, +\, \dfrac{x}{5}\, -\, \dfrac{y}{6}\, +\, ...\)

...with \(\displaystyle x,\, y\, >\, 0,\) for what values of \(\displaystyle x\) and \(\displaystyle y\) does this series converge conditionally, and for what values does it converge absolutely?

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I think I need to consider when x>y, x<y, and x=y. And I know that I need to use the alternating series test.
But I am still lost..

Any help will be appreciated!!

 

[tkhunny]

Given the following:

. . .\(\displaystyle x\, -\, \dfrac{y}{2}\, +\, \dfrac{x}{3}\, -\, \dfrac{y}{4}\, +\, \dfrac{x}{5}\, -\, \dfrac{y}{6}\, +\, ...\)

...with \(\displaystyle x,\, y\, >\, 0,\) for what values of \(\displaystyle x\) and \(\displaystyle y\) does this series converge conditionally, and for what values does it converge absolutely?

---

I think I need to consider when x>y, x<y, and x=y. And I know that I need to use the alternating series test.
But I am still lost..

Any help will be appreciated!!

Have you considered rewriting each successive pair of terms?

\(\displaystyle x - \dfrac{y}{2} = \dfrac{2x-y}{2}\)

\(\displaystyle \dfrac{x}{3} - \dfrac{y}{4} = \dfrac{4x-3y}{12}\)

\(\displaystyle \dfrac{x}{5} - \dfrac{y}{6} = \dfrac{6x-5y}{30}\)

...

\(\displaystyle \dfrac{x}{2k-1} - \dfrac{y}{2k} = \dfrac{(2k)x-(2k-1)y}{(2k-1)(2k)}\)


That might lead to something.