Question

Can anyone help me with this question? I don't know how to do it because of the ak

http://puu.sh/d7o8z/2f3fd4f8dc.png

Answer 1

@ganeshie8

Answer 2

Since the second series converges, the terms will converge to 0 :
\[\lim\limits_{k\to\infty} \dfrac{a_kk^2+2k}{{a_{k+1}} k ^2 + 5} = 0\]

yes ?

Answer 3

oo I see

Answer 4

which is same as

\[\lim\limits_{k\to\infty} \left|\dfrac{a_k}{a_{k+1}}\right| = 0\]

Answer 5

Next you can appeal to ratio test, just be careful the fraction is flipped ^

Answer 6

Ok understood! thanks so much for clearing that up! It is much appreciated!

Answer 7

np :) so does the first series converges or diverges ?

Answer 8

I believe it diverges

Answer 9

it has to diverge and believing it should be easy :) only justifying it is lil tricky

Answer 10

let \(\{a_k\}\) be a constant sequence, then the first series clearly doesn't converge.

the second series also doesn't converge because the limit of the terms wont converge to 0

that tells us that the second series doesn't converge when the sequence \(\{a_k\}\) is constant or decreasing.

Answer 11

so a necessary condition for second series to converge is increasign terms in \(\{a_k\}\)

In other words, a necessary condition for convergence of second series is the divergence of first series.

Answer 12

Gotcha! Thank you! Writing a testimonial for ya

Answer 13

wow! ty for the sweet testimonial :D