The Wednesday Morning Math Challenge: Week 8

Welcome to Week 8 of the Wednesday Morning Math Challenge. You can find solutions to the Week 7 puzzles here, along with — more importantly — discussions of some ways to approach them. Remember that the real goal here is to think creatively.

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This week, we are focusing on a single problem. It happens to be one of my favorite. The question goes as follows:

Consider the infinite sum of alternating numbers

1 – 1/2 + 1/3 – 1/4 + 1/5 – 1/6 + 1/7 – 1/8 + 1/9 – 1/10 + ….

This sum is known to equal the natural logarithm of two. Now consider the sum

1 – 1/2 – 1/4 + 1/3 – 1/6 – 1/8 + 1/5 – 1/10 – 1/12 + 1/7 – 1/14 – 1/16 + 1/9 – 1/18 – 1/20 + …..

Given what I have told you about the sum of the first series, what is the sum of the second?