Question

Check if the following infinite series is convergent or not

(1/2)+((1/3)*(1+1/2))+((1/4)*(1+1/2+1/3))+.....

Answer 1

can we create a summation rule for it?

Answer 2

or distribute it out and make multiple summations ....

Answer 3

we can write it as

sigma ((1/(n+1))*(1+1/2+1/3+.......1/n))

Answer 4

summation goes from 1 to infinity

Answer 5

\[1(\frac{1}2{}+\frac{1}{3}+\frac{1}{4}+...)\]
\[\frac12(\frac{1}{3}+\frac{1}{4}+\frac15+...)\]
\[\frac14(\frac{1}{4}+\frac15+\frac{1}{6}+...)\]
is what i think im seeing

Answer 6

yours might be more apt for a ratio test tho

Answer 7

yes but i am unable to write the (1+1/2+1/3+.....) term in short formula to apply that.

Answer 8

\[a_n=\frac{1}{2^{n+1}}\sum_{i=0}^{n}\frac{1}{(i+1)}\]

Answer 9

how did you write it?

Answer 10

same as yours, but im thinking if an starts at 1, then 2^(n) might be more appropriate

Answer 11

still not sure how well this will do for us .... the more basic idea is the distribution and collection of 'like' terms

Answer 12

(1/2)+((1/3)*(1+1/2))+((1/4)*(1+1/2+1/3))+.....

....


1(1/2 + 1/3 + 1/4 + 1/5 + 1/6 + ...)
1/2(1/3 + 1/4 + 1/5 + 1/6 + ...)
1/4(1/4 + 1/5 + 1/6 + ...)
1/8(1/5 + 1/6 + ...)

then the limit of a sum is the sum of the limits

Answer 13

i dont see that any of the parts converges .... the first part is the harmonic and doesnt converge ,,, i dont see the others converging either but i cant prove it at the moment

Answer 14

well actually this is a part of a bigger problem. We were supposed to check the convergence of the above series and another series just like above but with alternating terms which means :

(1/2)-((1/3)*(1+1/2))+((1/4)*(1+1/2+1/3))-.....

there are alternating signs with each term. This one i think can be done with liebnitz test.

Answer 15

im not familiar with a liebnitz test so i cant verify that

Answer 16

It is a test used to check the convergence of alternating infinite series. I spelled it wrong. It is "Leibnitz test".

Answer 17

the first one posted is similar to the harmonic series, but which some values doubled, tripled etc so its larger by defualt and diverges. no idea about the alternating one :)

Answer 18

right ,thanks