Question

The sum of the series
`(1)/(1!)x+(2)/(2!)x^(2)+(3)/(3!)x^(3)`….. To `infty` is
A. `e^(x)`
B. `x e^(x)`
C. `x e^(x)-1`
D. none of these

Answer 1

Answer:
We have
`1+(1)/(1!)x+(2)/(2!)x^(2)+(3)/(3!)x^(3)…infty`
`=underset(n=1)overset(infty)Sigma (n)/(n!)x^(n)`
`=underset(n=1)overset(infty)Sigma(1)/(n-1)!x^(n)=xunderset(n=1)overset(infty)Sigma(x^(n)-1)/(n-1)!=xe^(x)`