## badreferences 3 years ago Let $$n\in\mathbb N$$. For$e^xf_n(x)=\sum_{k=1}^\infty\frac{k^nx^k}{\left(k-1\right)!}$show that $$f_n(x)$$ is a polynomial of degree $$n+1$$ with integer coefficients. Tricky question.

@TuringTest @KingGeorge @Zarkon You guys might be interest.

@bahrom7893 You too, maybe lol.

3. TuringTest

I got a linear thingy when I tried it that makes no sense, I'll write my work in a minute just so somebody can laugh at it

4. bahrom7893

no im most likely not interested lol

5. bahrom7893

Anyway... I'm off for tonight guys, interviews in 10 hrs. I need my sleep. Gnite eastern front.

7. bahrom7893

thanks :)

8. perl

what is f sub n (x) ?

You can ignore the sub. It's just a marker to show that the function $$f$$ is dependent on $$n$$.

10. perl

well first lets look at e^x, whats the series of this

You don't have to walk me through it, lol, I already have the solution. This is just a very difficult challenge.

12. perl

e^x |dw:1348817638823:dw|

13. perl

so we can see immediately that

14. perl

|dw:1348817750800:dw|

15. perl

whats solution

I'll post it when I can pick it up, Sir, it's not in my possession right now.

17. perl

ohhh

18. perl

darn

19. perl

we know that

20. perl

|dw:1348817984611:dw|

21. mukushla

see if this is right or not http://openstudy.com/study#/updates/50651902e4b08d185211d536

22. mukushla

i proved that$f_1(x)=x+x^2$and$f_{n+1}(x)=x(f_n(x)+f_n^'(x))$