Question

how would you show this series converge
not using limit comparison test
\[\sum_{k=1}^{\infty}\frac{3\sqrt{k}+2}{2k^2+5}\]

Answer 1

we could compare it with \[\frac{1}{k^{3/2}}\]
but i want a systematic way rather than look for a p series that is greater and cvg

Answer 2

well im thinking if we simplify it to

k + 2 - 5k/sqrt(k)

does this work simpler?

Answer 3

5sqrt(k)/k^2 that is

Answer 4

hmm that would be good
but i don't where did you get that lol

Answer 5

multiply top and bottom by ksqrt(k) and then add zero to the top really

Answer 6

oh i see, i was just about thinking that is where you got it

Answer 7

the series that we get is \[\frac{5}{k^{3/4}}\]

Answer 8

oh yes 3/2, don't what is wrong with me today

Answer 9

\[\frac{k\sqrt k}{k\sqrt k}\frac{3\sqrt{k}+2}{(2k^2+5)}\]
\[\frac{3k^2+2k\sqrt k}{k\sqrt k(2k^2+5)}\]
\[\frac{3k^2-(k^2+k^2)+2k\sqrt k+(5-5)}{k\sqrt k(2k^2+5)}\]
\[\frac{2k^2+k^2+2k\sqrt k+5-5}{k\sqrt k(2k^2+5)}\]
\[\frac{(2k^2+5)+(k^2+2k\sqrt k-5)}{k\sqrt k(2k^2+5)}\]
hmm, seems i errored along the way

Answer 10

i actually did take \[\frac{3\sqrt{k}}{2k^2}=\frac{3}{2k^{3/2}}\]
i just did some wrong math and made it power 3/4 which gave something unwanted

Answer 11

:) bad math happens

Answer 12

darn it! too many bad math today!

Answer 13

it made think that the prof used something else lol

Answer 14

question
does harmonic series appear to converge to zero?
if we take out the first term a1=1

Answer 15

does not*

Answer 16

harmonic does not converge no matter how many discrete terms you remove.

infinity - 1 = infinity right?

Answer 17

yes true

Answer 18

unless, you remove specific terms only get 1 + 1/2 + 1/4 + 1/8 + ... :) and the like

Answer 19

well does not \[1/2+1/3+1/4...................\]
appear to be converging

Answer 20

the sequence 1/n converges, but a convergent sequence does not necessarily gives us a convergent sum of the terms of a sequence

Answer 21

or may be I am perceiving this the wrong way

Answer 22

oh yeah i gotcha you
i mixed up with sequence convergence :)

Answer 23

if the list of terms (the sequence) does NOT go to zero, then we cannot converge the sum

the converse of this is not true ..

Answer 24

i recall a youtube prof mentioned that it appear we will take forever to read the wall
when he talked about this series

Answer 25

to reach the wall*

Answer 26

1/2 = 1/2!
1/2 + 1/3 = 5/3!
5/6 + 1/4 = 21/4!
21/24 + 1/5 = 106/5!
106/5! + 1/6 = 636/6!

dunno if theres a pattern that we could determine in the partial sums that we could then limit out to show its not convergent

Answer 27

hmm seems it is growing to infinity but slowly?

Answer 28

yep