S-x-1-x-1-x-2-x-2-2-x-3-x-3-3-x-4-x-4-4-S-i-1-x-i-x-i-1-i-i-Does-S-limit-to-a-value-for-x-

Question Number 4454 by FilupSmith last updated on 29/Jan/16

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

S = \underset{i = 1}{\sum^{\infty}} \frac{x - i}{x^{i} - {(- 1)}^{i} i}

D oes S l i m i t t o a value for \pm x ?

Commented by prakash jain last updated on 30/Jan/16

∣ x ∣⩽ 1 \Rightarrow does not converge as \lim_{i \to \infty} a_{i} \neq 0

∣ x ∣> 1 \Rightarrow converges . Ratio test .

Answered by Jens last updated on 29/Jan/16

P r o b a b l y y e s i f ⌈ x] > 1 . D i v i d e t o p

a n d b o t t o m b y i . L e t i \Rightarrow \infty a n d t h e

n o m i n a t o r \Rightarrow - 1 w h i l e t h e

d e n o m i n a t o r \Rightarrow x^{i} / i \Rightarrow \infty s o e a c h

t e r m \Rightarrow 0 f a s t e r t h a n n e c e s s a r y

f o r c o n v e r g e n c e . I n f a c t \sum_{1}^{\infty} {i x}^{- i}

= - x \frac{d}{d x} (\frac{1}{x - 1}) = \frac{x}{{(x - 1)}^{2}} a s s u m i n g

∣ x ∣> 1 .

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