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(x-1)/(x+1) + (1/2)(x^2-1)/(x+1)^2 + (1/3)(x^3-1)/(x+1)^3+ ... infinity
![question image](http://images.topperlearning.com/topper/new-ate/811207log.png)
Asked by apoorv.bhardwaj100 | 30 Oct, 2022, 08:06: AM
![begin mathsize 14px style L e t space S space equals space fraction numerator x minus 1 over denominator x plus 1 end fraction plus 1 half fraction numerator x squared minus 1 over denominator left parenthesis x plus 1 right parenthesis squared end fraction space plus 1 third fraction numerator x cubed minus 1 over denominator left parenthesis x plus 1 right parenthesis cubed end fraction plus....................... t o space infinity end style](https://images.topperlearning.com/topper/tinymce/cache/fd3179a6984df236b20ec65f51423208.png)
Above series can be written as , S = S1 - S2
where ![begin mathsize 14px style space S subscript 1 space equals space fraction numerator x over denominator x plus 1 end fraction plus 1 half x squared over left parenthesis x plus 1 right parenthesis squared space plus 1 third x cubed over left parenthesis x plus 1 right parenthesis cubed plus....................... t o space infinity end style](https://images.topperlearning.com/topper/tinymce/cache/2097a44e4ab8b6f5c3f47935ca3441d6.png)
![begin mathsize 14px style space S subscript 1 space equals space fraction numerator x over denominator x plus 1 end fraction plus 1 half x squared over left parenthesis x plus 1 right parenthesis squared space plus 1 third x cubed over left parenthesis x plus 1 right parenthesis cubed plus....................... t o space infinity end style](https://images.topperlearning.com/topper/tinymce/cache/2097a44e4ab8b6f5c3f47935ca3441d6.png)
and ![begin mathsize 14px style space S subscript 2 space equals space fraction numerator 1 over denominator x plus 1 end fraction plus 1 half 1 over left parenthesis x plus 1 right parenthesis squared space plus 1 third 1 over left parenthesis x plus 1 right parenthesis cubed plus....................... t o space infinity end style](https://images.topperlearning.com/topper/tinymce/cache/368df17b6236cbf5cb2b02f88f45c9fe.png)
![begin mathsize 14px style space S subscript 2 space equals space fraction numerator 1 over denominator x plus 1 end fraction plus 1 half 1 over left parenthesis x plus 1 right parenthesis squared space plus 1 third 1 over left parenthesis x plus 1 right parenthesis cubed plus....................... t o space infinity end style](https://images.topperlearning.com/topper/tinymce/cache/368df17b6236cbf5cb2b02f88f45c9fe.png)
Let
and ![begin mathsize 14px style beta space equals space fraction numerator 1 over denominator x plus 1 end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/cd23bcabb9c7585df4ed8fd3987677af.png)
![begin mathsize 14px style alpha space equals space fraction numerator x over denominator x plus 1 end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/c027ce328ee8055ab4512453443ea602.png)
![begin mathsize 14px style beta space equals space fraction numerator 1 over denominator x plus 1 end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/cd23bcabb9c7585df4ed8fd3987677af.png)
Then we get
![begin mathsize 14px style S subscript 1 equals space 1 space plus space alpha space plus space alpha squared space plus space alpha cubed plus space....................... t o space infinity end style](https://images.topperlearning.com/topper/tinymce/cache/a95661849388aee905c665af15f1c67f.png)
![begin mathsize 14px style S subscript 2 equals space 1 space plus space beta space plus space beta squared space plus space beta cubed plus space....................... t o space infinity end style](https://images.topperlearning.com/topper/tinymce/cache/abc744a51c6370701682360fec3d8bae.png)
Sum of series S1 and S2 are written as
![begin mathsize 14px style S subscript 1 space equals space ln open parentheses fraction numerator 1 over denominator 1 minus alpha end fraction close parentheses space space space space space a n d space space space space S subscript 2 space equals space ln open parentheses fraction numerator 1 over denominator 1 minus beta end fraction close parentheses space end style](https://images.topperlearning.com/topper/tinymce/cache/38a05cc47081fe6fc44ea13d14fcc118.png)
![begin mathsize 14px style S subscript 1 space minus space S subscript 2 space equals space ln open parentheses fraction numerator 1 minus beta over denominator 1 minus alpha end fraction close parentheses space equals space ln open parentheses fraction numerator 1 minus begin display style fraction numerator 1 over denominator x plus 1 end fraction end style over denominator 1 minus begin display style fraction numerator x over denominator x plus 1 end fraction end style end fraction close parentheses space equals space ln left parenthesis x right parenthesis end style](https://images.topperlearning.com/topper/tinymce/cache/b4a55be029291072124762d06cd16988.png)
Answered by Thiyagarajan K | 20 Dec, 2022, 17:25: PM
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