exponential type series question

this looks really simple but might end up quite complicated. i'm currently stuck though :/

suppose that **for every k**, . does this imply that ?

the above condition can also be rewritten as of course, which makes it even more painfully obvious, but I still can't prove this...

I need also to figure out a more general question: does imply that , but clearly knowing the answer for the initial question would be a good start...

any partial ideas or references will be appreciated too!..