SOLUTION: Please sir , send me the value or method to find the value of the following summation. sum of ( a^n * b^n ) / (n !

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Question 42050: Please sir , send me the value or method to find the value of the following summation.


sum of ( a^n * b^n ) / (n ! )2

here a and b are non negative real constants and summation is over n where n varies 0 to infinity.
Answer by psbhowmick(878) About Me  (Show Source):
You can put this solution on YOUR website!
e%5E%28ax%29+=+1+%2B+ax%2F1%21+%2B+%28ax%29%5E2%2F2%21+%2B+%28ax%29%5E3%2F3%21+%2B+....%2B+%28ax%29%5En%2Fn%21+%2B+.....toinfinity
Similarly,
e%5E%28bx%29+=+1+%2B+bx%2F1%21+%2B+%28bx%29%5E2%2F2%21+%2B+%28bx%29%5E3%2F3%21+%2B+....%2B+%28bx%29%5En%2Fn%21+%2B+.....toinfinity

Multiplying (1) and (2) we have
e%5E%28ax%29%2Ae%5E%28bx%29+ = (1+%2B+ax%2F1%21+%2B+%28ax%29%5E2%2F2%21+%2B+%28ax%29%5E3%2F3%21+%2B+....%2B+%28ax%29%5En%2Fn%21+%2B+.....toinfinity) (1+%2B+bx%2F1%21+%2B+%28bx%29%5E2%2F2%21+%2B+%28bx%29%5E3%2F3%21+%2B+....%2B+%28bx%29%5En%2Fn%21+%2B+.....toinfinity)

or e%5E%28%28a%2Bb%29x%29+ = (1+%2B+ax%2F1%21+%2B+%28ax%29%5E2%2F2%21+%2B+%28ax%29%5E3%2F3%21+%2B+....%2B+%28ax%29%5En%2Fn%21+%2B+.....toinfinity) (1+%2B+bx%2F1%21+%2B+%28bx%29%5E2%2F2%21+%2B+%28bx%29%5E3%2F3%21+%2B+....%2B+%28bx%29%5En%2Fn%21+%2B+.....toinfinity)

Now, let us find the coefficient of x%5E%282n%29 in left side.
This is
[%2B+%28a%5En%2Fn%21%29%28b%5En%2Fn%21%29+%2B.....+%2B+%28a%5E%282n%29%2F%282n%29%21%29]

Again, coefficient of x%5E%282n%29 in the expansion of e%5E%28%28a%2Bb%29x%29 is %28a%2Bb%29%5E2n%2F%282n%29%21.
Therefore we can write
%28a%2Bb%29%5E%282n%29%2F%282n%29%21+ = %2B+%28a%5En%2Fn%21%29%28b%5En%2Fn%21%29+%2B.....+%2B+%28a%5E%282n%29%2F%282n%29%21%29


Perhaps this is the expression you want.