October 22nd, 2018, 08:11 AM  #1 
Senior Member Joined: Dec 2015 From: somewhere Posts: 606 Thanks: 88  Convergence Series
Show whether the series converges or diverges $\displaystyle \sum \limits_{n=1}^{\infty} \left(\frac{1+3+3^2+...+3^n}{3^n + 3^{n1} }\right)^n$ Last edited by skipjack; October 22nd, 2018 at 12:38 PM. 
October 22nd, 2018, 09:12 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,533 Thanks: 1390 
$\displaystyle \lim \limits_{n\to\infty}~\frac{\sum \limits_{k=0}^n~3^k}{3^n+3^{n1}} = \frac 9 8 > 1$ So the root test says this series diverges. Last edited by skipjack; October 23rd, 2018 at 04:24 PM. 
October 22nd, 2018, 01:29 PM  #3 
Global Moderator Joined: May 2007 Posts: 6,807 Thanks: 717 
The fraction is always > 1, so there is no way for the series to converge.

October 22nd, 2018, 11:56 PM  #4  
Member Joined: Oct 2018 From: Netherlands Posts: 39 Thanks: 3  Quote:
Appendum: I suppose that's the outcome of ROMSEKs post as well, but I couldn't read it since I got a dfracerror message. Last edited by Arisktotle; October 23rd, 2018 at 12:37 AM.  
October 23rd, 2018, 12:26 AM  #5 
Global Moderator Joined: Dec 2006 Posts: 20,942 Thanks: 2210 
I've changed it to use frac instead of dfrac.

October 23rd, 2018, 12:40 AM  #6 
Member Joined: Oct 2018 From: Netherlands Posts: 39 Thanks: 3 
@skipjack. Thanks, but 1 dfrac instance left in romsek's post. I get these messages regularly. Anything I can do to overcome them? Last edited by skipjack; October 23rd, 2018 at 04:30 PM. 
October 23rd, 2018, 01:20 PM  #7 
Global Moderator Joined: May 2007 Posts: 6,807 Thanks: 717  The question was about the series, not the individual terms.
Last edited by skipjack; October 23rd, 2018 at 04:31 PM. 

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convergence, serie, series 
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