division explanation

**l flipboi l** · September 23rd 2011, 05:41 PM

Hello,

I was wondering... Why is it divided by $n^2$ ? This is actually a calculus problem, trying to determine if the series is convergent or divergent.

But, I just needed some clarification on the algebra.

Thanks,

**Soroban** · September 23rd 2011, 07:03 PM

Hello, l flipboi l!

Why is it divided by $n^2$ ?

$\lim_{n\to\infty}\frac{n}{\sqrt{10+n}} \;=\;\lim_{n\to\infty}\dfrac{1}{\sqrt{\dfrac{10}{n ^2}+\dfrac{1}{n}}} \;=\;\infty$

We have the fraction: . $\frac{n}{\sqrt{10+n}}$

$\text{Divide top and bottom by }n\!:$

. . $\dfrac{\dfrac{n}{n}}{\dfrac{\sqrt{10+n}}{n}} \;=\;\dfrac{1}{\dfrac{\sqrt{10+n}}{\sqrt{n^2}}} \;=\;\frac{1}{\sqrt{\dfrac{10+n}{n^2}}} \;=\;\frac{1}{\sqrt{\dfrac{10}{n^2} + \dfrac{n}{n^2}}} \;=\;\frac{1}{\sqrt{\dfrac{10}{n^2} + \dfrac{1}{n}}}$

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