real analysis - Show that $Nsum_{n = 2}^{infty} frac{t^n}{nN^n}$ converges uniformly to $0$

I want to show that $$N\sum_{n = 2}^{\infty} \frac{t^n}{nN^n}$$ converges uniformly to $0$ as $N \to \infty$ on a bounded interval.

I have done a very few problems on showing that the power series converges uniformly on a bounded interval using Weierstrass M-test. But these problems have something in common that it just asks to show "uniform convergence" to whatever, not to a specific value such as $0$ in this problem.

I cannot find a nice definition of uniform convergence of series. But series is just a special type of function, right?

So given $\epsilon >0$, I have to show an $N_0 \in \mathbb{N}$ such that $N \geq N_0$ implies $|S_N(t) - 0| < \epsilon,$ where $$S_N(t) = N\sum_{k = 2}^{\infty} \frac{t^k}{kN^k},$$ for $N \in \mathbb{N}$.

But I cannot proceed further.