February 26th, 2018, 12:53 PM  #1 
Newbie Joined: Aug 2017 From: Norway Posts: 10 Thanks: 0  Series
Hey! I'm stuck on this task about series. Determine whether the following series converges or diverges: $\displaystyle \sum\limits_{n=0}^\infty \frac{n}{n^2 \sqrt n +1}$ I've tried to use the ratio test. All help much appreciated. Last edited by deltaX; February 26th, 2018 at 01:00 PM. 
February 26th, 2018, 01:39 PM  #2 
Math Team Joined: Jul 2011 From: Texas Posts: 2,755 Thanks: 1405 
note $\displaystyle \sum_{n=0}^\infty \dfrac{n}{n^2\sqrt{n}+1} = \sum_{n=1}^\infty \dfrac{n}{n^2\sqrt{n}+1}$ further ... $\dfrac{n}{n^2\sqrt{n}+1} = \dfrac{1}{n\sqrt{n}+\frac{1}{n}} < \dfrac{1}{n^{3/2}}$ note $\displaystyle \sum_{n=1}^\infty \dfrac{1}{n^{3/2}}$ is a convergent pseries 
February 26th, 2018, 02:08 PM  #3  
Senior Member Joined: Oct 2009 Posts: 406 Thanks: 141  Quote:
Note that if your series is a fraction of terms in n to some (possibly fractional) power, then a comparison will always work. The limit comparison test is there probably the easiest.  

Tags 
series 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Using series manipulations to find the sums of finite series  stevewilliams  Algebra  1  April 22nd, 2016 01:40 PM 
Is finding laurent series expansion of f at z_0 using geometric series convenient?  king.oslo  Complex Analysis  0  December 28th, 2014 06:50 AM 
In need of help disk, series test, taylor, and power series  g0bearmon  Real Analysis  2  May 22nd, 2012 12:10 PM 
Convergent series > series of geometric means converges  The Chaz  Real Analysis  11  February 7th, 2011 04:52 AM 
In need of help disk, series test, taylor, and power series  g0bearmon  Calculus  1  December 31st, 1969 04:00 PM 