Prove that:

$\displaystyle \frac{(1+\frac{1}{n})^x}{1+\frac{x}{n}}=1+\frac{x( x-1)}{2n^2}+o\left(\frac{1}{n^2}\right)$

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- Aug 17th 2011, 10:05 PMAlso sprach ZarathustraLittle o notation.
Prove that:

$\displaystyle \frac{(1+\frac{1}{n})^x}{1+\frac{x}{n}}=1+\frac{x( x-1)}{2n^2}+o\left(\frac{1}{n^2}\right)$ - Aug 18th 2011, 06:00 AMemakarovRe: Little o notation.
I believe that $\displaystyle (1+y)^\alpha=1+\alpha y+\alpha(\alpha-1)y^2/2+o(y^2)$ (perhaps the $\displaystyle o(y^2)$ part needs to be double-checked). Expanding the nominator and the denominator in this way gives you the right-hand side.

- Aug 18th 2011, 06:13 AMCaptainBlackRe: Little o notation.
You are interested in the rate of growth of the left hand side as $\displaystyle n \to \infty$ so put $\displaystyle \varepsilon=1/n$ and expand as a Mclaurin series with the remainder term after the term in $\displaystyle \varepsilon^2$. Changing back to $\displaystyle n$ will give the result.

CB - Aug 18th 2011, 08:37 AMAlso sprach ZarathustraRe: Little o notation.