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Question

If f=x1+x2+13(x1+x2)3+15(x1+x2)5+... and g=x23x3+15x5+17x7+29x9+..., then f=13cg. Find c.

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Solution

Let y=x1+x2
then f=y+y33+y55+...
=12log(1+y1y)
=12log⎜ ⎜1+x1+x21x1+x2⎟ ⎟
=12log(1+x+x21x+x2) ...(1)
and g=x23x3+15x5+17x7+29x9+111x11+113x13+215x15+...

=x(113)x3+15x5+17x713(113)x9+111x11+113x1315(113)x15+...
=(x+x33+x55+x77+...)(x3+x93+x155+...)
=12log(1+x1x)12log(1+x31x3)
=12log[(1+x)(1x).(1+x3)(1x3)]
=12log[(1+x)(1x)(1+x+x2)(1x)(1+x)(1x+x2)]
=12log(1+x+x21x+x2)
from (1) and (2), we get
fg
c=3

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