Calculating the tail of the Zeta function

The well known Zeta function is this:

I found this function:

as A get's larger this function get's closer and closer to the true value. For instance:

whereas the sum on the left by itself is only:

So the integral appears to be adding very close to the right amount. 

and making A larger...

The integral on the right actually has a definite form (for each value k):

where the Ci function is:

And the gamma constant is:

Now back to the original formula I found:

There is a simple reason why it works... the integral of cos(pi*x)^2 from an integer minus .5 to that integer plus .5 gets closer and closer to 1/2 for larger integers. So a function you multiply it by is made like a discrete sum over the integers divided by 2. It is actually a handy way to convert from a discrete sum to an integral that I'll probably go more into in the future. 

The example for k=3...

So the integral is adding something around .02 and getting a lot of decimal places correct.