Convergence of Sum of a series $$\frac{(1+r)^{N+1}-(1+r)-rN}{r^2(1+r)^N}$$
I'm calculating the sum of a special series and finally get a general formula for the series. $r$ is a constant in the formula. And I'm curious whether this series will be convergent or not. Could someone explain how to calculate the sum when $N$ is towards infinity? Thank you very much.

Welcome to MSE!
If you write $\alpha = 1+r$ to simplify notation you get
$$
\frac{1}{(\alpha-1)^2}
\left (
\alpha - \frac{\alpha}{\alpha^n} - \frac{\alpha n}{\alpha^n} + \frac{n}{\alpha^n}
\right )
$$
But we know that

*

*$\frac{\alpha}{\alpha^n} \to 0$ when $|\alpha| > 1$

*$\frac{\alpha n}{\alpha^n} \to 0$ when $|\alpha| > 1$

*$\frac{n}{\alpha^n} \to 0$ when $|\alpha| > 1$
This is basically because $\alpha^n$ grows exponentially in $n$, which dominates any linear (or constant) growth. You can use L'hospital if you want to verify this yourself.
Then, applying these limits, we see that your limit converges if and only if $|\alpha| > 1$, and then it converges to
$$\frac{\alpha}{(\alpha - 1)^2}$$
I'll leave it to you to convert this back into a formula in terms of $r$.
If you like, you can code this up quite quickly in
sage and convince yourself that this formula is correct. Checking with $n \approx 500$ should be good enough to see the convergence for most values of $\alpha$.

I hope this helps ^_^
