Let $f_0(x)=e^x $ and $f_{n+1} (x)=xf_n'(x) $ for $n=0,1,2,... $ ........Find the value of$$\sum_{n=0}^{\infty} \frac{f_n(1)}{n!}$$

Question

Let  $f_0(x)=e^x $  and  $f_{n+1} (x)=xf_n'(x) $ for  $n=0,1,2,... $ ........Find the value of$$\sum_{n=0}^{\infty} \frac{f_n(1)}{n!}$$

dominusscholae · Answer

e^2?

anonymous · Answer

that was close...nope...

experimentx · Answer

the coefficients are 
                                    exp(1)
                                    exp(1)
                                   2 exp(1)
                                   5 exp(1)
                                   15 exp(1)
                                   52 exp(1)
                                  203 exp(1)
                                  877 exp(1)
                                  4140 exp(1)
                                 21147 exp(1)
                                 115975 exp(1)

anonymous · Answer

$$e^x=\sum_{k=0}^{\infty} \frac{x^k}{k!}$$

dominusscholae · Answer

Here's my work so far.|dw:1343730839276:dw|