Let x=1+a+a^2+.... and y=1+b+b^2+..... , where |a|<1 and |b|<1. Prove that

1+ab+a^2b^2+......=xy/(x+y-1)

Asked by Vineeth K | 21st Mar, 2015, 08:23: PM

Expert Answer:

$G i v e n space t h a t space x equals 1 plus a plus a squared plus.... space a n d space y equals 1 plus b plus b squared plus... comma space w h e r e space open vertical bar a close vertical bar less than 1 space a n d space open vertical bar b close vertical bar less than 1. W e space n e e d space t o space p r o v e space t h a t 1 plus a b plus a squared b squared plus... equals fraction numerator x y over denominator x plus y minus 1 end fraction N o w space c o n s i d e r space x colon space x equals 1 plus a plus a squared plus.... R e w r i t i n g space t h e space a b o v e space e q u a t i o n comma space w e space h a v e comma x equals open parentheses 1 minus a close parentheses to the power of negative 1 end exponent space space space space space space space space space open square brackets b y space b i n o m i a l space f o r m u l a close square brackets rightwards double arrow x equals fraction numerator 1 over denominator 1 minus a end fraction S i m i l a r l y comma space y equals fraction numerator 1 over denominator 1 minus b end fraction C o n s i d e r space x plus y minus 1 x plus y minus 1 equals fraction numerator 1 over denominator 1 minus a end fraction plus fraction numerator 1 over denominator 1 minus b end fraction minus 1 rightwards double arrow x plus y minus 1 equals fraction numerator 1 minus b plus 1 minus a minus open parentheses 1 minus a close parentheses open parentheses 1 minus b close parentheses over denominator open parentheses 1 minus a close parentheses open parentheses 1 minus b close parentheses end fraction rightwards double arrow x plus y minus 1 equals fraction numerator 1 minus b plus 1 minus a minus open parentheses 1 minus b minus a plus a b close parentheses over denominator open parentheses 1 minus a close parentheses open parentheses 1 minus b close parentheses end fraction rightwards double arrow x plus y minus 1 equals fraction numerator 1 minus b plus 1 minus a minus 1 plus b plus a minus a b over denominator open parentheses 1 minus a close parentheses open parentheses 1 minus b close parentheses end fraction rightwards double arrow x plus y minus 1 equals fraction numerator 1 minus a b over denominator open parentheses 1 minus a close parentheses open parentheses 1 minus b close parentheses end fraction A n d comma space w e space h a v e comma x y equals open parentheses fraction numerator 1 over denominator 1 minus a end fraction close parentheses open parentheses fraction numerator 1 over denominator 1 minus b end fraction close parentheses rightwards double arrow x y equals fraction numerator 1 over denominator open parentheses 1 minus a close parentheses open parentheses 1 minus b close parentheses end fraction N o w space c o n s i d e r space R H S colon R H S equals fraction numerator x y over denominator x plus y minus 1 end fraction equals fraction numerator fraction numerator 1 over denominator open parentheses 1 minus a close parentheses open parentheses 1 minus b close parentheses end fraction over denominator fraction numerator 1 minus a b over denominator open parentheses 1 minus a close parentheses open parentheses 1 minus b close parentheses end fraction end fraction equals fraction numerator 1 over denominator 1 minus a b end fraction equals open parentheses 1 minus a b close parentheses to the power of negative 1 end exponent equals 1 plus a b plus a squared b squared plus... equals L H S H e n c e space p r o v e d.$

Answered by Vimala Ramamurthy | 24th Mar, 2015, 08:49: AM

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