Question

If `y=1+x+(x^2/(2!))+(x^3/(3!))+(x^4/(4!))+..... oo` so prove that ? `(dy)/(dx) = y`

If `y=1+x+(x^2/(2!))+(x^3/(3!))+(x^4/(4!))+..... oo` so prove that `(dy)/(dx) = y`

Answer 1

Please see below.

Explanation:

Here,

`y=1+x+x^2/(2!)+x^3/(3!)+x^4/(4!)+x^5/(5!)+...oo`

`(dy)/(dx)=0+1+(2x)/(2!)+(3x^2)/(3!)+(4x^3)/(4!)+(5x^4)/(5!)+...oo`

#(dy)/(dx)=1+(2x)/(2xx1!)+(3x^2)/(3xx2!)+(4x^3)/(4xx3!)+ (5x^4)/(5xx4!)+...oo#

`(dy)/(dx)=1+x+x^2/(2!)+x^3/(3!)+x^4/(4!)+...oo`

`(dy)/(dx)=y`

Note :

`(i)1! =1`

`(ii)2! =2xx1!`

`(iii)3! =3xx2!`

`(iv)4! =4xx3!`
...............
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`n! =nxx(n-1)!`