# Thread: determine whether the sequence converrger or diverges

1. ## determine whether the sequence converrger or diverges

Here is the problem:
an = n/(1+sqrt(n))
I divided top and bottom by n
then had 1 in the numerator
and 1/n + 1/n^(1/2) in the denominator
because there is still an n in the denominator
so the sequences diverges, is this correct?
The problems with n^2 or n seem so easy but this one where it is sqrt(n) or N^(1/2) gives me trouble.

Thank you,
Keith

2. Hello, Keith!

$\displaystyle a_n \:= \:\frac{n}{1+\sqrt{n}}$

Divide top and bottom by $\displaystyle \sqrt{n}$ .**

We have: .$\displaystyle \frac{\dfrac{n}{\sqrt{n}}\quad} {\dfrac{1}{\sqrt{n}} + \dfrac{\sqrt{n}}{\sqrt{n}}} \;=\;\frac{\sqrt{n}}{\dfrac{1}{\sqrt{n}} + 1}$

Then: .$\displaystyle \lim_{n\to\infty}\left(\frac{\sqrt{n}}{\frac{1}{\s qrt{n}} + 1}\right) \;=\;\frac{\infty}{0 + 1} \;=\;\infty$ . . . diverges

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**

Divide through by the highest power of the variable in the denominator.