Convergence of an alternating series

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Main Question or Discussion Point

Consider a sequence with the ##n^{th}## term ##u_n##. Let ##S_{2m}## be the sum of the ##2m## terms starting from ##u_N## for some ##N\geq1##.

If ##\lim_{N\rightarrow\infty}S_{2m}=0## for all ##m##, then the series converges. Why?

This is not explained in the following proof:
Screen Shot 2016-06-11 at 1.20.39 am.png

Screen Shot 2016-06-11 at 1.20.51 am.png
 
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Answers and Replies

  • #2
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Consider a sequence with the ##n^{th}## term ##u_n##. Let ##S_{2m}## be the sum of the ##2m## terms starting from ##u_N## for some ##N\geq1##.

If ##\lim_{N\rightarrow\infty}S_{2m}=0## for all ##m##, then the series converges. Why?
This is not true, consider ##u_n=(-1)^n##. You also have to add that un are alternating and decreasing in magnitude.

The proof looks a bit sloppy, but you can use a similar approach to show that the partial sum is always between two numbers that approach the same limit. Sandwich theorem.
 

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