Bound for Fourier Sum

Question

Let $t\in \mathbb{R} \;\text{and}\; \lambda =2\alpha(2+ \alpha^{-1})$ then $$\|f-f_t\|_2 ^2 =\sum_{n \in \mathbb{Z}}|c_n(f)(1-e^{-int})|^2.$$ and suppose there exist $ \alpha >0$ such that $$\sum_{|n| >N}|c_n(f)|^2 \leq(N+1)^{-\alpha}.$$ Show that there exist constant $K \in \mathbb{R}$ such that $$\|f-f_t\|_2 ^2 \leq K\cdot t^{\lambda}$$ whenever $0<t<1$. Here $$c_n(f)= \frac{1}{2\pi} \int_{[-\pi,\pi]}f(x)e^{-inx}dx.$$

My attempt: Let $\beta=2(2+ \alpha^{-1})$. And let $0<t<1$. Choose $N \in \mathbb{N}$ such that $N\leq t^{-\beta}<N+1$. I know i should estimate the sum $\sum_{|n| \leq N}$. But from here i dont know what to do!! Any help and hints would be much appreciated.