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Question

If f=x1+x2+13(x1+x2)3+15(x1+x2)5+... to and g=x23x3+15x5+17x729x9+..., then f=d×g. Find 4d.

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Solution

f=x1+x2+13(x1+x2)3+15(x1+x2)5+...
We know, Log(1+x)=xx22+x33x44...

f=12[log(1+x1+x2)log(1x1+x2)]
=12[log(1+x+x21+x2)log(1+x2x1+x2)]
f=12log(x2+x+1x2x+1)
Now, g=x23x3+15x5+17x729x9+...
g=x(113)x3+15x5+17x713(113)x9+...
g=(x+13x3+15x5+17x7+......)(x3+(x3)33+(x3)53+.....)
We know, 12(Log(1+x)Log(1x))=x+x33+x55....
g=12[log(1+x)log(1x)]12[log(1+x3)log(1x3)]
=12log(1+x1x)12log(1+x31x3)
g=12log((1+x)(1x3)(1x)(1+x3))
g=12log(x2+x+1x2x+1)
g=f

Comparing with given value ,
d=1
4d=4

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