Question

Prove this identity between two infinite sums (with x ∈ R and n! stands for factorial):

Answer 1

Answer:

( n=0∑∞n!x n ) 2 =( n=0∑∞n!x n )( m=0∑∞m!x m )=k=0∑∞m+n=k

m,n∑n!x nm!x m =

=\sum\limits_{k=0}^{\infty}\sum\limits_{n=0}^k\frac{x^n}{n!}\frac{x^{k-n}}{(k-n)!}=\sum\limits_{k=0}^{\infty}\left(\sum\limits_{n=0}^k\frac{1}{n!}\frac{1}{(k-n)!}\right)x^k==

k=0∑∞

n=0∑kn!x n(k−n)!x k−n = k=0∑∞

( n=0∑kn!1(k−n)!1 )x k=

=\sum\limits_{k=0}^{\infty}\left(\sum\limits_{n=0}^k1^n\binom{k}{n}1^{k-n}\right)\frac{x^k}{k!}=\sum\limits_{k=0}^{\infty}(1+1)^k\frac{x^k}{k!}=\sum\limits_{k=0}^{\infty}\frac{(2x)^k}{k!}=

k=0∑∞ ( n=0∑k1 n ( nk )1 k−n ) k!x k = k=0∑∞

(1+1) kk!x k= k=0∑∞k!(2x) k

its the answer........