June 1st, 2017, 07:25 PM  #1 
Member Joined: Jan 2017 From: California Posts: 80 Thanks: 8  sequence convergence
Hi guys, is it safe to assume that this sequence $$a_{n}=\sqrt{\frac{125n^3+7n}{5n^3+7n^2}}$$ diverges since the nth term test leads to a limit equal to 5? Thanks. Last edited by skipjack; June 1st, 2017 at 10:47 PM. 
June 1st, 2017, 07:28 PM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,094 Thanks: 2360 Math Focus: Mainly analysis and algebra 
Well, if the terms tend to 5, doesn't that mean that the sequence converges (to 5)? The series $\sum a_n$ would diverge though. 
June 1st, 2017, 07:50 PM  #3 
Member Joined: Jan 2017 From: California Posts: 80 Thanks: 8  The limit of the nth term is equal to 5. I thought the series diverges anytime that limit is different from 0. That's why I concluded that it diverged.
Last edited by skipjack; June 1st, 2017 at 10:48 PM. 
June 1st, 2017, 07:54 PM  #4 
Member Joined: Jan 2017 From: California Posts: 80 Thanks: 8  Ok, I just understood what you meant... thanks again.
Last edited by skipjack; June 1st, 2017 at 10:49 PM. 
June 1st, 2017, 09:40 PM  #5 
Senior Member Joined: Aug 2012 Posts: 1,662 Thanks: 427  In your OP, you called it a sequence. That's just the list of terms. The series is the sum. Two different things. Remember: The sum of the series is defined as the limit of the sequence of partial sums.
Last edited by skipjack; June 1st, 2017 at 10:50 PM. 

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