Little o notation.

• 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!