is not the sum of the infinite series, but the partial sum .
You see it in the definition of the sum of an infinite series is defined as , where is as above.
the partial sums are:
It's no longer infinite, just the first n terms.
So you can see that the 1/3 in the first term cancels the one in the third term, the 1/4 in the second term cancels the one in the fourth term, and so on. But when we get to 1/(n+1), there is no term to cancel it, and the same for 1/(n+2).
Hope this helps - feel free to post again in this thread if you're still having trouble.