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- Thread starter dwsmith
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- #1

- Jan 26, 2012

- 890

\[ \left( \frac{1}{\sqrt{n - 1}} - \frac{1}{\sqrt{n}} \right) \sim \frac{n^{-3/2}}{2}\]$\sum\limits_{n = 2}^{\infty}n^p\left(\frac{1}{\sqrt{n - 1}} - \frac{1}{\sqrt{n}}\right)$

where p is any fixed real number.

If this was just the telescoping series or the p-series, this wouldn't be a problem.

CB

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How did you come up with that though?\[ \left( \frac{1}{\sqrt{n - 1}} - \frac{1}{\sqrt{n}} \right) \sim \frac{n^{-3/2}}{2}\]

CB

- Feb 13, 2012

- 1,704

Kind regards

$\chi$ $\sigma$

- Jan 26, 2012

- 890

\[\begin{aligned}\frac{1}{\sqrt{n-1}}-\frac{1}{\sqrt{n}}&=\frac{1}{\sqrt{n}\sqrt{1-1/n}}-\frac{1}{\sqrt{n}}\\&= \frac{1}{\sqrt{n}}\left(1+(-1/2)(-n)^{-1}+O(n^{-2})\right)-\frac{1}{\sqrt{n}}\\ & = \dots \end{aligned}\]How did you come up with that though?

CB