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Prove that if the series $x_{n}$ converges then series $(x_{m}+x_{k})$ converges, where $k=2n+1$ and $m=2n$. So if i show that the partial sum of $x_{n}$ is convergent how does this have to compare with the partial sum of the other series?? Do I have to use the limit definition for the partial sums? Also is there an example where the converse is invalid? (only hint is enough for this part!) Please help!

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Write out the first few terms of each. From what I can tell they're the same series, just being written two terms at a time in the second case. –  Robert Mastragostino Oct 28 '12 at 15:39
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Hint about the converse: try x(2n)+x(2n+1)=0 for every n. –  Did Oct 28 '12 at 15:39
    
that sounds like an alternating series right?? –  d13 Oct 28 '12 at 15:41
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Alternating series has a very specific meaning (alternation in sign, plus $|a_{n+1}|\le |a_n|$ plus $\lim a_n=0$. You will want your terms to alternate in sign, but likely want the $|a_n|$ not to get small. –  André Nicolas Oct 28 '12 at 16:32

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The problem here says:

Suppose $x_0 + x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 + x_ 9 + \cdots$ converges.

Prove: $a_0 + a_1 + a_2 + a_3 + a_4 + \cdots$ converges, where $a_n = x_{2n} + x_{2n + 1}$.

(For example, $a_3 = x_6 + x_7$.)

In particular, the $n$th partial sum of the $a_n$ series is the $(2n+1)$st partial sum of the $x_n$ series.

So: yes, you can use the "limit definition for partial sums" (isn't this how you defined convergence of an infinite series?) to prove that the $a_n$ series converges.

For the converse, let $x_n = (-1)^n$.

Does $\sum x_n$ converge? Does $\sum a_n$ converge?

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thankyou very much! i appreciate ur help :)) –  d13 Oct 29 '12 at 5:56

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