Little o notation.

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August 17th 2011, 10:05 PM
Also sprach Zarathustra

Little o notation.

Prove that:

$\frac{(1+\frac{1}{n})^x}{1+\frac{x}{n}}=1+\frac{x( x-1)}{2n^2}+o\left(\frac{1}{n^2}\right)$
August 18th 2011, 06:00 AM
emakarov

Re: Little o notation.

I believe that $(1+y)^\alpha=1+\alpha y+\alpha(\alpha-1)y^2/2+o(y^2)$ (perhaps the $o(y^2)$ part needs to be double-checked). Expanding the nominator and the denominator in this way gives you the right-hand side.
August 18th 2011, 06:13 AM
CaptainBlack

Re: Little o notation.

Quote:

Originally Posted by Also sprach Zarathustra

Prove that:

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

You are interested in the rate of growth of the left hand side as $n \to \infty$ so put $\varepsilon=1/n$ and expand as a Mclaurin series with the remainder term after the term in $\varepsilon^2$ . Changing back to $n$ will give the result.

CB
August 18th 2011, 08:37 AM
Also sprach Zarathustra

Re: Little o notation.

Quote:

Originally Posted by CaptainBlack

You are interested in the rate of growth of the left hand side as $n \to \infty$ so put $\varepsilon=1/n$ and expand as a Mclaurin series with the remainder term after the term in $\varepsilon^2$ . Changing back to $n$ will give the result.

CB

It worked! Thank you!

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