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Question

Sum of infinity the following series:
1+4x2+7x4+10x6+...,|x|<1.

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Solution

S=1+4x2+7x4+10x6+... ..................(1)
Sx2=x2+4x4+7x6+.... ...............(2)

eqn(1)eqn(2), we get
(1x2)S=1+3x2+3x4+3x6+... .........(3)

This is a G.P. of common ratio =x2

So, using sum of G.P.=a(1r)
Here, a=first term and r=common ratio

So,
(1x2)S=1+3x2(1x2)

(1x2)S=1x2+3x2(1x2)

S=1+2x2(1x2)2

Hence, this is the answer.

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