Kenneth Rosen Edition 6th Exercise 7.4 Question 4g (Page No. 496)

Question

Find a closed form of the generating function of the following sequence

$0, 1, -2, 4, -8, 16, -32,64,.....$

Answer 1

Standard Generating function defined  as :

$G(x)=a_{0}x^{0}+a_{1}x^{1}+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}+a_{5}x^{5}+a_{6}x^{6}+........$

$G(x)=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}+a_{5}x^{5}+a_{6}x^{6}+........$--------------->$(1)$

where $a_{0},a_{1},a_{2},a_{2},a_{3},a_{4},a_{5},..........$ are generating sequence

Given that generating sequence are $0,1,−2,4,−8,16,−32,64,.....$

$G(x)=0+1.x-2x^{2}+4x^{3}-8x^{4}+16x^{5}-32x^{6}+64x^{7}-..........$    $[$From the equation$(1)]$

$G(x)=x - 2x^2 + 2^2x^3 -2^3x^4 + 2^4x^5 - 2^5x^6 + 2^6x^7-........$----------->$(2)$

This is a infinite GP where $a = x$ and $r = -2x$

$\large S_\infty = \frac{a}{1-r} =$$\Large   \frac{x}{1+2x}$

So$,G(x)=\frac{x}{1+2x}$

Comments

one correction in place of 2r it should be 2x in last line