- #1
bennyska
- 112
- 0
Homework Statement
first off, my latex isn't coming out just right. I've attached a pdf that should look right. if you can fill in the gaps from what I've posted here, sweet, otherwise, if you wouldn't mind checking out the attached pdf, that would be awesome.
[itex]Assume that $\sum_{k=1}^{\infty}a_{k}$ converges with each $a_{k} >0. $
Let $s_{n}=\sum_{k=1}^{\infty}a_{k}.$ Show that $\sum_{k=1}^{\infty}a_{k}/s_{k}$
converges.
Homework Equations
The Attempt at a Solution
[itex]Assume that $\sum_{k=1}^{\infty}a_{k}$ converges with each $a_{k} >0. $
Let $s_{n}=\sum_{k=1}^{\infty}a_{k}.$ Show that $\sum_{k=1}^{\infty}a_{k}/s_{k}$
converges.
Proof: Since $\sum_{k=1}^{\infty}a_{k}$ converges, we have that $(s_{n})_{n=1}^{\infty}$
converges.
We use the ratio test. Let $b_{k}=a_{k}/s_{k}.$ Then $\lim_{k\rightarrow\infty}b_{k+1}/b_{k}=\lim_{k\rightarrow\infty}\dfrac{\dfrac{a_{k+1}}{s_{k+1}}}{\dfrac{a_{k}}{s_{k}}}=\lim_{k\rightarrow\infty}\dfrac{a_{k+1}s_{k}}{a_{k}s_{k+1}}=\lim_{k\rightarrow\infty}\dfrac{a_{k+1}}{a_{k}}\cdot\lim_{k\rightarrow\infty}\dfrac{s_{k}}{s_{k+1}}$
By the ratio test, since $\sum_{k=1}^{\infty}a_{k}$ converges, $\lim_{k\rightarrow\infty}\dfrac{a_{k+1}}{a_{k}}=L,$
for some $L<1.$
Also, since $(s_{n})_{n=1}^{\infty}$ converges, we have that $\lim_{k\rightarrow\infty}\dfrac{s_{k}}{s_{k+1}}=1.$
Then $\lim_{k\rightarrow\infty}\dfrac{a_{k+1}}{a_{k}}\cdot\lim_{k\rightarrow\infty}\dfrac{s_{k}}{s_{k+1}}=M$
for some $M<1,$ so by the ratio test, $\sum_{k=1}^{\infty}a_{k}/s_{k}$
converges.[/itex]
possible problems with my proof (of which there are at least a few that i am aware of)
1: ratio test is not an "if and only if", so i don't know i can make the conclusion i did that lim ak+1/ak < 1.
2: i never really used the fact that ak > 0 for all k. I'm tempted to say that because of that fact, sk/sk+1 < 1 for all k, so multiplying that by lim ak+1/ak, which is also less than 1 results in a number < 1.
3: i didn't really show that lim sk/sk+1 = 1. it seems intuitive to me that it is, but without having anything actually defined, I'm sort of at a loss to actually prove it.
any guidance would be greatly appreciated.