Solve-the-following-sum-i-1-1-i-1-ix-i-x-2x-2-3x-3-4x-4-

Question Number 2258 by Filup last updated on 12/Nov/15

Solve the following sum :

\underset{i = 1}{\sum^{\infty}} {(- 1)}^{i + 1} {i x}^{i} = x - 2 x^{2} + 3 x^{3} - 4 x^{4} + \dots

Commented by Filup last updated on 12/Nov/15

For : ∣ x ∣< 1

Answered by prakash jain last updated on 12/Nov/15

S = x - 2 x^{2} + 3 x^{3} - 4 x^{4} + \dots (1)

x S = + x^{2} - 2 x^{3} + 3 x^{4} - \dots . (2)

add (1) and (2)

(1 + x) S = x - x^{2} + x^{3} - x^{4} + \dots = \frac{x}{1 + x}

S = \frac{x}{{(1 + x)}^{2}}

Commented by Filup last updated on 12/Nov/15

Wow! I couldnt work this out!

I^{'} ll have to remember that!

Commented by 123456 last updated on 14/Nov/15

x = 1

1 - 2 + 3 - 4 + \cdot \cdot \cdot = \frac{1}{{(1 + 1)}^{2}} = \frac{1}{4} XD

Commented by prakash jain last updated on 14/Nov/15

1 - 1 + 1 - 1 + 1 - 1 + \dots . = \frac{1}{2} ?