f-n-x-x-1-x-2x-2-x-nx-n-x-n-N-f-n-x-f-n-1-x-nx-n-x-f-1-x-x-1-x-f-n-1-x-f-n-x-nx-n-x-n-Z-f-0-x-f-1-x-x-1-x

Question Number 5585 by 123456 last updated on 21/May/16

f_{n} (x) = \frac{x}{1 - x} + \frac{2 x}{2 - x} + \dots + \frac{n x}{n - x}

n \in N^{*}

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f_{n} (x) = f_{n - 1} (x) + \frac{n x}{n - x}

f_{1} (x) = \frac{x}{1 - x}

f_{n - 1} (x) = f_{n} (x) - \frac{n x}{n - x}

n \in Z

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f_{0} (x) = f_{1} (x) - \frac{x}{1 - x} = 0

f_{- 1} (x) = f_{0} (x) = 0

f_{- 2} (x) = f_{- 1} (x) - \frac{x}{1 + x} = - \frac{x}{1 + x}

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f_{- n} (x) = - \frac{x}{x + 1} - \frac{2 x}{x + 2} - \dots - \frac{(n - 1) x}{x + (n - 1)}

n \in Z_{-}^{*}, n < - 1

Commented by Rasheed Soomro last updated on 21/May/16

What is required ?

Commented by 123456 last updated on 21/May/16

\frac{{d f}_{n}}{d x}

Commented by FilupSmith last updated on 22/May/16

f_{n} (x) = \frac{x}{1 - x} + \frac{2 x}{2 - x} + \dots + \frac{n x}{n - x}

f_{n} (x) = \underset{i = 1}{\sum^{n}} \frac{i x}{i - x}

\frac{{d f}_{n}}{d x} = \frac{d}{d x} \underset{i = 1}{\sum^{n}} \frac{i x}{i - x}

\frac{{d f}_{n}}{d x} = \underset{i = 1}{\sum^{n}} \frac{i (i - x) + i x}{{(i - x)}^{2}} q u o t a n t r u l e

\frac{{d f}_{n}}{d x} = \underset{i = 1}{\sum^{n}} \frac{i^{2}}{{(i - x)}^{2}}

? ? ? ?