prove:
1 - 1/2 + 1/3 - 1/4 + 1/5 + ... + (-1)^n (1/n)
is always positive
The basic reasons why are the following:
The partial sums of an even number of terms is positive always because an even partial sum is the previous even partial sum, plus a number, minus a number (e.g. +1/3, -1/4); and the term you're subtracting is smaller than the term you're adding, so the partial sum must increase. In other words, the next even partial sum is always bigger than the last. So if the first even partial sum is positive (which we can check it is), all the even partial sums are positive.
Now we only need to show the odd partial sums are also all positive. This actually follows from what we just argued (that the even partial sums are all positive), provided that the first partial sum is positive too. This is because an odd partial sum is the previous even partial sum plus a positive number (e.g. the terms up to -1/4, PLUS 1/5). So the odd partial sums are all greater than the last even partial sum, so they're all positive too.
Also, everything we said applies to any sum of the form a
1 - a
2 + a
3 - a
4 + ... + (-1)
n+1 a
n, where the a
j are all positive and strictly decreasing.
