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!
Comparing series with convergent series
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sequences-and-series
convergence-divergence
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1*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. – 2012-10-28
1 Answers
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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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0thankyou very much! i appreciate ur help :)) – 2012-10-29