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"If "0ltxlt1," prove that "(1)/(1+x)+(2x...

`"If "0ltxlt1," prove that "(1)/(1+x)+(2x)/(1+x^(2))+(4x^(3))/(1+x^(4))+...oo=(1)/(1-x)`

Text Solution

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The given series is in the form
`(f'_(1)(x))/(f_(1)(x))+(f'_(2)(x))/(f_(2)(x))+(f'_(3)(x))/(f_(3)(x))+...oo`
Then consider the product `f_(1)(x)cdotf_(2)(x)cdotf_(3)(x)...f_(n)(x)." Now"`
`(1-x)(1+x)(1+x^(2))(1+x^(4))...(1+x^(2^(n-1)))" (1)"`
`=(1-x^(2))(1+x^(2))(1+x^(4))...(1+x^(2^(n-1)))`
`=(1-x^(4))(1+x^(4))...(1+x^(2^(n-1)))` :
`=(1-x^(2^(n-1)))(1+x^(2^(n-1)))`
`1-x^(2^(n))`
`"Now, when "nrarroo,x^(2^(n-1))rarr0(because0ltxlt1).`
`"Therefore, taking "nrarroo," in (1), we get"`
`(1-x)(1+x)(1+x^(2))(1+x^(4))...=1`
Taking logarithm, we get
`log(1-x)+log(1+x)+log(1+x^(2))+log(1+x^(4))+...=0`
Differentiating w.r.t.x, we get
`-(1)/(1-x)+(1)/(1+x)+(2x)/(1+x^(2))+(4x^(3))/(1+x^(4))+...=0`
`"or "(1)/(1+x)+(2x)/(1+x^(2))+(4x^(3))/(1+x^(4))+...oo=(1)/(1-x)`
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