Help with proving properties of Tails of Series

**zachoon** · December 2nd 2012, 06:23 PM

Hi, I have an issues with solving this problem.

Let the summation of a_n from n=1 to infinity be a series and let m be in the natural numbers. Show that the summation of a_n from n=1 to infinity converges IFF the summation of a_m+n from n=1 to infinity converges, and that (the summation of a_n from n=1 to infinity) = S_m + (the summation of a_m+n from n=1 to infinity converges).

I understand that this problem is basically telling you to prove the properties of the Tails of series, but I'm not sure how I would go about proving it. And sorry for all the long typing for all the summation things, I don't know how to type it out the proper way.

**jakncoke** · December 2nd 2012, 06:41 PM

Well, $\displaystyle\sum_{n=1}^{\infty} a_n = \displaystyle\sum_{n=1}^{m} a_n +\displaystyle\sum_{n=1}^{\infty} a_{n+m}$

Now if $\displaystyle\sum_{n=1}^{\infty} a_n = L$ , Then $\displaystyle\sum_{n=1}^{m} a_n +\displaystyle\sum_{n=1}^{\infty} a_{n+m} = L$ .
Since $\displaystyle\sum_{n=1}^{\m} a_n = k$ where k is a finite number(you are just adding up a finite number of terms so you get a finite number).
$\displaystyle\sum_{n=1}^{\infty} a_{n+m} = L - K (a finite term)$ . So $\displaystyle\sum_{n=1}^{\infty} a_{n+m}$ converges.

It should be pretty straightforward.

**zachoon** · December 2nd 2012, 06:52 PM

Hi thanks for responding. However, it says also to prove (the summation of an from n=1 to infinity) = Sm + (the summation of am+n from n=1 to infinity converges), which is basically what you wrote for the first line, so I feel like the answer you provided is incomplete. And sorry if this is a misguided response, but I'm still very lost in the whole field of real analysis still. so sorry and thank you.

**jakncoke** · December 2nd 2012, 06:56 PM

$\displaystyle\sum_{n=1}^{m} a_n = a_1 + a_2 + ... + a_m$
$\displaystyle\sum_{n=1}^{\infty} a_{n+m} = a_{m+1} + a_{m+2} + ...$

Now i don't know how to be more clear than that, surely adding those two u get $a_1 + a_2 + .... + a_m + a_{m+1} + .... . = \displaystyle\sum_{n=1}^{\infty} a_n$

**zachoon** · December 2nd 2012, 07:07 PM

Okay, Thank you. I get it now

.

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