Infinite sum of a series

I have to find out how the sum from 1 to N of a series scales, as N tends to infinity. I have tried approximating it with an integral but this doesn't seem to work. What other methods could I use to approach this problem? It's also not a typical series, for example geometric series, so I cannot just substitute the formula in for that.

You can't come up with a general rule for how series scale since, as an example, a gp with r<1 sums to infinity bu if r>=1 then it won't.

Some series will converge and others won't so a particular series or type of series needs to be looked at separately.

Can you expand on "I need to know how it goes to infinity"?

What type of characteristics are you looking for confirmation of?

The exact value of the series at each and every N is not important so let me try and make this clear.
If I have this series
$\sum_n^N f(n,N)$
If we let N tend to infinity, we need to know how fast the series "grows"
$\sum_n^N f(n,N) = cN^2$
As N tends to infinity, the coefficient c determines the value of the series for any N. This coefficient however is not important to me, what's important is the "N^2", or whatever it's supposed to be, as this shows how slow/fast the series scales to infinity. I hope this is clear enough.

It is still not very clear to me, since you did not answer my specific question, and just added more confusing notation (what is c?).

You can do something with integrals, but you need to be careful.



Essentially, you need to find two functions, one of which you can prove grows at least as fast as your series, and one which grows at most as fast, which grow at the same rate. Think about how you might want to shift your integrals to make this work (drawing pictures might help).

In my case the series isn't as simple and an integral approximation doesn't work very well or at all it seems. I think that an integral approximation only works well in some cases for series and in this case it fails. This is why I am looking for other methods.

Should work fine for 1/sqrt(n). In general, the method that I'm thinking of will work for $\sum f(n)$ whenever $f : \mathbb{R} \to \mathbb{R}$ is convex (I think, I haven't actually proved that, but it seems right, and it certainly works in this case). As a hint, your series is the integral of a step function. How might you approximate that step function above and below with smooth curves?

I really don't think the series we're discussing in this case will work with integrals. If you want I can PM you the series I'm working with so you can see, because I'd rather not write it on the thread if possible. Just let me know, and thanks for the help.

Sorry if I wasn't clear in my first post. I am expecting that sum to go to infinity as N tends to infinity, but I need to know how it goes to infinity, it's not enough to say that it goes to infinity. I can give the series if I really have to but I would rather not, and want to know if there are any other feasible approaches to this problem.