Thursday, 29 August 2013

sum of the series frac12x1x+x2+frac4x32x1x2+x4+frac8x74x31x4+x8+cdotscdots




If |x|<1, Then the sum of the series 2x11x+x2+4x32x1x2+x4+8x74x31x4+x8+




Try: Let A=12x1x+x2+4x32x1x2+x4+8x74x31x4+x8+




Adx



=[12x1x+x2+4x32x1x2+x4+8x74x31x4+x8+]dx



Adx=ln[(1x+x2)(1x2+x4)(1x4+x8)]



Now i have seems that expression under ln on



Right side must have closed form in $-1




But i could not understand how can i find it.



could some help me, thanks


Answer



Hint:



(1+x+x2)(1x+x2)=(1+x2)2x2=?



nr=1(1x2r+(x2r)2)=1x2n+1+(x2n+1)21+x+x2




Now limn2n+1



and |x|<1,limmxm=0


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