Series convergence test... double square root?

Nov 2018
2
0
california
Hi guys, new to the forum. I had this question on a test in Calculus, and it really stumped me.

\(\displaystyle [FONT=&amp]\sum[/FONT][FONT=&amp]_{n=2}^{\infty}\frac{1}{\sqrt{n*\sqrt{n-1}}}[/FONT][\math]

I don't know if i wrote that correctly, but it's essentially a series with a denominator consisting of sqrt(n(sqrt(n-1))), starting at 2 and going to infinity. The instructions for the question is to determine whether the series converges or diverges, and to state the method used. I tried splitting the denominator into n's to the powers of -1/2 and -1/4, then using the partial fractions. But my attempt was incorrect.

Does anyone have any idea how to approach this problem? I really thought I was onto something with the fractional exponents.

(Another image, just in case.)


(EDIT: yup, totally messed up the latex code. Sorry about that.)\)
 

romsek

MHF Helper
Nov 2013
6,828
3,073
California
Just compare this series with the harmonic series.

$n>1 \Rightarrow \dfrac{1}{\sqrt{n\sqrt{n-1}}} > \dfrac 1 n$

$\sum \dfrac 1 n$ is known to diverge so this series must as well.
 
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Nov 2018
2
0
california
Thank you for the explanation! Turns out I was thinking too much about it after all. :)