# Understanding this proof involving alternating series

1. Aug 28, 2013

### chipotleaway

I'm having trouble the underlined red part of this proof (attached image) of the what looks to be the alternate series test, not sure if it's an error but it's more likely I've perhaps misunderstood something.

If $y_j$ is defined as the sequence of partial sums of the even terms of the sequence $x_j$ from j=n onwards (i.e. the positive terms), then shouldn't $y_{j+1}=y_j + x_{N+2j+2}$?

How does $x_{N+2j+1}$ come in? Thats an odd/negative term of $x_N$!
And then the result is that $y_j \geq y_{j+1}$, but but if [itex]y_j[/tex] is the sequence of positive partial sums, then should it not be increasing?

#### Attached Files:

• ###### proof.JPG
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2. Aug 28, 2013

### eigenperson

yj is not the sum of the even terms only. It's the sum of the 2j+1 terms from xN through xN+2j.