Given a(n) a series that is monotonic and converges to zero which means: a(n)---->0.
How to prove that the following summation converges for any K (a normal number):
$$\sum\limits_{n=1}^\infty (-1)^{[n/k]}a(n)$$
Note: [] symbol rounds the value down.
What did I do?
First, I have two options:
1) a(n) is a monotonically increasing function. Thus a(n)>=0
2) a(n) is a monotonically decreasing function. Thus a(n)<=0
But from here I couldn't continue since all the laws of convergence which I know, only work for positive summation or by proving that the summation in || converges which proves that the original converges but that doesn't seem to be the case here -i.e take 1/n as an example of a(n)-