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 October 22nd, 2018, 08:11 AM #1 Senior Member   Joined: Dec 2015 From: somewhere Posts: 734 Thanks: 98 Convergence Series Show whether the series converges or diverges $\displaystyle \sum \limits_{n=1}^{\infty} \left(\frac{1+3+3^2+...+3^n}{3^n + 3^{n-1} }\right)^n$ 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: 2,586 Thanks: 1430 $\displaystyle \lim \limits_{n\to\infty}~\frac{\sum \limits_{k=0}^n~3^k}{3^n+3^{n-1}} = \frac 9 8 > 1$ So the root test says this series diverges. Thanks from SenatorArmstrong, idontknow and ProofOfALifetime Last edited by skipjack; October 23rd, 2018 at 04:24 PM.
 October 22nd, 2018, 01:29 PM #3 Global Moderator   Joined: May 2007 Posts: 6,835 Thanks: 733 The fraction is always > 1, so there is no way for the series to converge. Thanks from SenatorArmstrong and ProofOfALifetime
October 22nd, 2018, 11:56 PM   #4
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Quote:
 Originally Posted by mathman The fraction is always > 1, so there is no way for the series to converge.
In fact, the fraction itself converges to 27/24 which is absolutely >1.

Appendum: I suppose that's the outcome of ROMSEKs post as well, but I couldn't read it since I got a dfrac-error 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: 21,034 Thanks: 2271 I've changed it to use frac instead of dfrac. Thanks from greg1313
 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
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Quote:
 Originally Posted by Arisktotle In fact, the fraction itself converges to 27/24 which is absolutely >1. Appendum: I suppose that's the outcome of romsek's post as well, but I couldn't read it since I got a dfrac-error message.
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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