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Prove that $$\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\cdots \cdots \cdots -\frac{1}{99}+\frac{1}{100}>0.2$$

$\bf{My\; Try::}$ We can write series as $$\frac{1}{2}\bigg(1+\frac{1}{2}+\frac{1}{3}+\cdots \cdots +\frac{1}{50}\bigg)-\bigg(\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\cdots \cdots +\frac{1}{99}\bigg)$$

Now How can i solve after that, Help Required, Thanks

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    $\begingroup$ Do you mean 0.2 or 0.5? $\endgroup$ – Henning Makholm Nov 20 '16 at 10:42
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    $\begingroup$ @HenningMakholm As the LHS is $\approx 0.31$, I supposeit should be $0.2$ $\endgroup$ – Hagen von Eitzen Nov 20 '16 at 10:43
  • $\begingroup$ Might be easier to rewrite the LHS to $\frac 22 H_{50} - H_{100} +1 $. $\endgroup$ – Henning Makholm Nov 20 '16 at 10:44
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    $\begingroup$ I can't understand why the OP doesn't answer direct questions asked to him. -1 $\endgroup$ – DonAntonio Nov 20 '16 at 10:49
  • $\begingroup$ To DonAntonio actually it does not strike in my mind. $\endgroup$ – juantheron Nov 20 '16 at 10:50
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$$\left(\frac12-\frac13\right)+\left(\frac14-\frac15\right)+\cdots+\left(\frac1{98}-\frac1{99}\right)+\frac1{100}>\left(\frac12-\frac13\right)+\left(\frac14-\frac15\right)=0.21\overline 6$$

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  • $\begingroup$ (+1) I was just about to give a very similar answer. I hope you don't mind my edit. $\endgroup$ – robjohn Nov 20 '16 at 10:49
  • $\begingroup$ Yup, you beat me to it +1. Darn time difference (5 am in Chgo). $\endgroup$ – Oscar Lanzi Nov 20 '16 at 10:59

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