Calculus II - Series and Convergence

[GreenPrint]

Homework Statement

Determine if the series

inf
Sigma n/(2n+1)
n=1

converges

Homework Equations

The Attempt at a Solution

When i did this I originally I thought I would just apply the divergence test

lim n/(2n+1) =/= 0
n->inf

there fore I thought by the divergence test the series diverges but I guess sense the limit is defined to be 1/2 it converges...

I'm confused... I think I may be over complicating this but by the divergence test shouldn't this series diverge sense the limit does not equal zero?

Thanks for any help

 

[stringy]

... but by the divergence test shouldn't this series diverge sense the limit does not equal zero?

Yes, that's correct. It's more commonly called the nth term test for divergence.

 

[gb7nash]

Homework Statement

Determine if the series

inf
Sigma n/(2n+1)
n=1

converges

Homework Equations

The Attempt at a Solution

When i did this I originally I thought I would just apply the divergence test

lim n/(2n+1) =/= 0
n->inf

there fore I thought by the divergence test the series diverges but I guess sense the limit is defined to be 1/2 it converges...

I'm confused... I think I may be over complicating this but by the divergence test shouldn't this series diverge sense the limit does not equal zero?
Thanks for any help

The bolded part is correct. Since you took the limit of the summand and received 1/2, the limit diverges. If the sum converges its limit is equal to 0.

However, the converse of the statement is not true. Don't fall into the trap of thinking the sum converges if the limit is equal to 0. If you obtain a limit of 0, you need to resort to a different method to determine if the sum converges.

 

[GreenPrint]

hm thanks i think i got it mixed up and it asked me if the sequence converges which it does lol

 

[stringy]

hm thanks i think i got it mixed up and it asked me if the sequence converges which it does lol

No it doesn't. It diverges. You just showed this using your test. Or was that a typo?

And it's a series, not a sequence.

 

[GreenPrint]

ya i got it and understand now i got a different part mixed up and stuff