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Question

The value of 13+232+333+434+ is

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Solution

L=13+232+333+434+
L3=132+233+334+435+2L3=13+132+133+134+2L3=1/3113
L=34=0.75

Alternate Solution:

We know that
11x=n=0xn , if |x|<1
Differentiate both the sides w.r.t. x, we get
1(1x)2=n=0nxn1
1(1x)2=n=1nxn1
(as first term is zero at n=0)
Now, put x=13
1(113)2=n=1n3n1
94=3n=1n3n
n=1n3n=34=0.75




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