Finding limit of function using series expansion

**thepopasmurf** · March 26th 2010, 08:19 AM

Hi,

I'm trying to prove the following limit

$\sqrt{x(x+1)}-x = \frac{1}{2}$ as x approaches infinity.

I'm getting stuck on the expansion of
$x\sqrt{1+1/x}$
I know that it's supposed to be
$x(1+\frac{1}{2x}-\frac{1}{8x^2} +.....)$
but I can't work out the series.
Can anyone help?

**Laurent** · March 26th 2010, 08:31 AM

Originally Posted by thepopasmurf

Hi,

I'm trying to prove the following limit

$\sqrt{x(x+1)}-x = \frac{1}{2}$ as x approaches infinity.

I'm getting stuck on the expansion of
$x\sqrt{1+1/x}$
I know that it's supposed to be
$x(1+\frac{1}{2x}-\frac{1}{8x^2} +.....)$
but I can't work out the series.
Can anyone help?

You don't need a series expansion, but an asymptotic expansion : $\sqrt{1+u}=1+\frac 12 u+o(u)$ when $u\to 0$ . Then, substituting, $\sqrt{x(x+1)}-x=x(\sqrt{1+\frac{1}{x}}-1)=x(1+\frac{1}{2x}+o(\frac{1}{x})-1)=\frac{1}{2}+o(1)$ , which means that the limit is $\frac{1}{2}$ .

**thepopasmurf** · March 26th 2010, 09:01 AM

Oh I see. Thanks. But why does it not work when I try to evaluate the expansion directly?

**Laurent** · March 26th 2010, 10:32 AM

Originally Posted by thepopasmurf

Oh I see. Thanks. But why does it not work when I try to evaluate the expansion directly?

You mean write the series expansion, for $x>1$ , $\sqrt{x^2+x}-x=x\left(\sqrt{1+\frac{1}{x}}-1\right)$ $=x\left(1+\frac{1}{2x}+\sum_{k=2}^\infty a_k\frac{1}{x^k} -1\right)= \frac{1}{2}+\sum_{k=2}^\infty \frac{a_k}{x^{k-1}}$ where $a_k=\frac{\frac{1}{2}(\frac{1}{2}-1)\cdots (\frac{1}{2}-k+1)}{k!}$ ? Then you have to justify that the last series converges to 0 when $x\to+\infty$ . There are ways to do that (this is an alternate series so we can get a uniform bound for the remainder of the series by Leibniz theorem), but it is not trivial, and first of all the asymptotic expansion is so much more appropriate (it is dedicated to computation of limits).

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