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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

Asked by apoorv.bhardwaj100 | 30 Oct, 2022, 08:06: AM

Expert Answer

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

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

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

Let

begin mathsize 14px style alpha space equals space fraction numerator x over denominator x plus 1 end fraction end style

and

begin mathsize 14px style beta space equals space fraction numerator 1 over denominator x plus 1 end fraction end style

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

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

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

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

Answered by Thiyagarajan K | 20 Dec, 2022, 17:25: PM

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