Find-the-sum-of-n-terms-of-the-following-series-1-1-x-1-1-x-1-1-x-

Question Number 6329 by Rasheed Soomro last updated on 24/Jun/16

Find the sum of n terms of the following series

\frac{1}{1 + \sqrt{x}} + \frac{1}{1 - x} + \frac{1}{1 - \sqrt{x}} + \dots .

Commented by FilupSmith last updated on 24/Jun/16

s equence x, \sqrt{x}, x \dots can be given by :

\frac{x}{2} (1 - {(- 1)}^{n + 1}) + \frac{\sqrt{x}}{2} (1 - {(- 1)}^{n})

1, 0, 1, 0, \dots is given by :

\frac{1}{2} (1 - {(- 1)}^{n})

S = \underset{n = 1}{\sum^{n}} (\frac{1}{1 + \frac{1}{2} (x (1 - {(- 1)}^{n + 1}) + \sqrt{x} (1 - {(- 1)}^{n}))})

Possibly a simpler method by using an

exponential notation .

i . e . S = Σ \frac{1}{1 + x^{a_{n}}}

where a_{n} toggles between 1 and 1 / 2

EDIT :

Another form :

S = \underset{n = 1}{\sum^{n}} \frac{1}{1 + x^{\frac{1}{2} ((1 - {(- 1)}^{n + 1}) + \frac{1}{2} (1 - {(- 1)}^{n}))}}

Commented by FilupSmith last updated on 24/Jun/16

another form :

S = \underset{n = 1}{\sum^{n}} \frac{1}{1 + x^{\frac{1}{2} (1 - {(- 1)}^{n + 1})} + x^{\frac{1}{4} (1 - {(- 1)}^{n})}}

Commented by Rasheed Soomro last updated on 24/Jun/16

T h e q u e s t i o n i s f r o m a t e x t b o o k

o f m a t h e m a t i c s a n d t h e r e a r e

o n l y t h r e e t e r m s g i v e n .

Commented by Yozzii last updated on 24/Jun/16

A r e t h o s e t h e o n l y f i r s t t e r m s g i v e n b y

t h e q u e s t i o n ?

p a t t e r n o n \sqrt{x} \to - x \to - \sqrt{x} \dots ?

Commented by prakash jain last updated on 24/Jun/16

\frac{1}{1 - x} - \frac{1}{1 + \sqrt{x}} = \frac{1 - (1 - \sqrt{x})}{1 - x} = \frac{\sqrt{x}}{1 - x}

\frac{1}{1 - \sqrt{x}} - \frac{1}{1 - x} = \frac{1 + \sqrt{x} - 1}{1 - x} = \frac{\sqrt{x}}{1 - x}

The given series an AP with common difference

\frac{\sqrt{x}}{1 - x}