I've been doing a lot of stuff with infinite series and Python lately, and have been getting some very interesting results. For example, the series 1 + 1/16 - 1/256 - 1/4096 + ... (two plus signs, then two minus signs, etc.) converges to 272/257 - wow! But the series 1 + 1/16 + 1/256 + 1/4096 - ... (four alternating plus and minus signs) converges to 69904/65537 - a very interesting result indeed.
Now I want to look at a series that doesn't involve powers of two. For example, I already know that the series 1 + 1/2 + 1/3 + 1/4 + ... does not converge, but the series 1 - 1/2 + 1/3 - 1/4 + ... converges to ln(2), and that's not too hard to prove.
But what about the series 1 + 1/2 - 1/3 - 1/4 + ... ? That has two alternating plus and minus signs. When I ran 1,000,000 iterations of the sequence in Python, I obtained a result approximately equal to 1.131971. I've checked a few things, and it's not ln(3), ln(4), ln(pi), or anything else (it's about ln(3.1017), but as far as I am concerned, the number 3.1017 does not have any significance here). Can someone please help me figure this out? (I don't have any experience with calculus)
Also, I tried a series involving triangular numbers. The series 1 + 1/3 + 1/6 + 1/10 + ... does not seem to converge. What about the series 1 - 1/3 + 1/6 - 1/10 + ... ? There HAS to be some sort of special result to that one, and after 10,000,000 iterations the result stabilized around 0.77258872.