Let $\lambda\in (0,1)$.
Is it true that for all $\epsilon>0$ there exists $N\in \mathbb{N}$ such that for all $(a_n)_{n\in\mathbb{N}}\in[0,1]^{\mathbb{N}}$ $$\bigg\vert\sum_{n=0}^{+\infty} \lambda (1-\lambda)^n a_n - \sum_{n=0}^{N} \lambda (1-\lambda)^n a_n\bigg\vert< \epsilon.$$