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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) + ...`
Solution
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)` = ey
⇒ 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!) .....`
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