Kenneth Rosen Edition 6th Exercise 6.4 Question 8 c (Page No. 440)

Question

Provide close formula for the sequences it determines

$\dfrac{1}{1−2x^2}$

Answer 1

$\Large \frac{1}{1-ay} = \sum_{k=0}^{\infty} a^k y^k = 1 + ay + a^2 y^2 + ...$

$\large \frac{1}{1-2x^2}$=$1+(2x^2)+(2x^2)^2+(2x^2)^3+(2x^2)^4........$

              =$1+2x^2+4x^4+8x^6+16x^8+........$

           closed form of above series -  we can see $a_n=0$, when n is odd

                            $a_n=2^{n/2}$,when n is even.