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Beloved community,

Does the following series converge?

$$\sum_{n=0}^\infty \left(\sqrt[n]{n} - \sqrt[n+1]{n+1}\right)$$

According to Wolfram Alpha, it does by the Comparison Test. However, after thinking about it long and hard, I still haven't found any series to compare it to.

Many thanks in advance! :)

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    $\begingroup$ It's a telescoping series. $\endgroup$ – carmichael561 Mar 30 '16 at 21:51
  • $\begingroup$ @carmichael561 gee whizz, I should have written out a few terms; it's fairly obvious. thanks $\endgroup$ – Mitch Baker Mar 30 '16 at 21:56
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$$\sum_{k=1}^n\left(\sqrt[k]k-\sqrt[k+1]{k+1}\right)=1-\sqrt2+\sqrt2-\sqrt[3]3+\sqrt[3]3-\sqrt[4]4+\ldots+\sqrt[n]n-\sqrt[n+1]{n+1}=$$

$$=1-\sqrt[n+1]{n+1}\xrightarrow[n\to\infty]{}0$$

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    $\begingroup$ This is perfect. thanks. from now on, I will write out the first few terms. lol. $\endgroup$ – Mitch Baker Mar 30 '16 at 21:58
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    $\begingroup$ @MitchBaker You're welcome, and great idea. $\endgroup$ – DonAntonio Mar 30 '16 at 21:58

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