Re: Why is this series convergent?

On Feb 25, 1:41 pm, "TCL" <t...@xxxxxxxxxxx> wrote:
1-(1/2)-(1/3)+(1/4)+(1/5)+(1/6)-(1/7)-(1/8)-(1/9)-(1/10)+(1/11)+.........
where the number of signs increases by one in each block.
I read somewhere that this series converges. Why?

Consider the partial sums S_n = 1 - 1/2 + 1/3 + ... - (-1)^n/n

If you list just the odd sums, you get: S_1 = 1, S_3 = 5/6, S_5 =
47/60, etc. This is strictly decreasing.

Now consider the even sums: S_2 = 1/2, S_4 = 7/12, S_6 = 37/60, etc.
This is strictly increasing.

But any two consecutive partial sums differ by magintude (-1)^n/n. So
both sequences must converge to the same value.

This argument can be applied to any alternating sequence with terms
that tend to zero.

.