My Math Forum sequence convergence

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 June 1st, 2017, 06:25 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 09:47 PM.
 June 1st, 2017, 06:28 PM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,616 Thanks: 2605 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. Thanks from dthiaw
June 1st, 2017, 06:50 PM   #3
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Quote:
 Originally Posted by v8archie 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.
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 09:48 PM.

June 1st, 2017, 06:54 PM   #4
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Quote:
 Originally Posted by v8archie 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.
Ok, I just understood what you meant... thanks again.

Last edited by skipjack; June 1st, 2017 at 09:49 PM.

June 1st, 2017, 08:40 PM   #5
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 Originally Posted by dthiaw Ok, I just understood what you meant... thanks again.
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 09:50 PM.

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