Question

If y = `x + x^2/2 + x^3/3 + x^4/4  ...`, then show that x = `y - y^2/(2!) + y^3/(3!) - y^4/(4) + ...`

Answer 1

y = `x + x^2/2 + x^3/3 + x^4/4  ...`

(i.e) y = `-[-x - x^2/2 - x^3/3 - x^4/4 ...]`

= – log(1 –  x)

(i.e) y = – log(1 –  x) 

= `log  1/(1 - x)`

So `log  1/(1 - x)` = y

⇒ `1/(1 - x)` = e^y

⇒ 1 – x = `1/"e"^y`

= e^–y

⇒ 1 – x = e^–y

⇒ 1 – e^–y

^{= x}

(i.e) x = `1 - [1 - y + y^2/(2!) - y^3/(3) + y^4/(4!) ...]`

= `1 - 1 + y - y^2/(2) + y^3/(3!) - y^4/(4!)`

(i.e) x = `y - y^3/(2!) + y^3/(3!) - y^4/(4!) .....`