$sup_{x\in R} |\sum λ_n e^{-|x-n|}|\leq \frac{e+1}{e-1}$

Question

Let $(λ_n)$ be a sequence convergent to zero. If $|λ_{n}|\leq 1$ for each $n\in \mathbb{N} $ then $sup_{x\in \mathbb{R}} |\sum λ_n e^{-|x-n|}|\leq \frac{e+1}{e-1}$ Could you give me a suggestion to face this problem?.

Answer 1

Hint: we can show that $\sum e^{-|x-n|} \leq \frac {e+1} {e-1}$ and this implies the given inequality. Split the sum into sum over $n >x$ and $n \leq x$ and you will get two geometric series. Once you write down the sum of these two series it is not hard to verify the inequality.