Ι got a feeling that $$\sum_{x=1}^{\infty}\Big\lvert\sum_{k=0}^{\infty} \frac{x^{2k}}{(k+1)!}(-1)^{k} \Big\rvert \geq \sum_{n=1}^{\infty} \frac{1}{n} $$
because $$\sum_{x=1}^{\infty} \Big\lvert x-\frac{x^2}{3!}+\frac{x^4}{5!}... \Big\rvert \geq 1+\frac{1}{2}+\frac{1}{3}...$$ i feel that somehow terms will get canceled but i cant prove it!!
**this came up as a part of problem i was solving . I got no idea if the above inequallity is true **
