Question

The coefficient of $x^n,nϵN$, in the expansion of $\frac{1+x}{1+x^2}$ is

• A. ${(-1)}^{n/2}$, when $n$ is an even integer
• B. $0$, when $n$ is an odd integer
• C. $1$, when $n$ is an even integer
• D. ${(-1)}^{(n-1)/2}$, when $n$ is an odd integer

Answer 1

Answer: (A), (D)

${(-1)}^{n/2}$, when $n$ is an even integer
${(-1)}^{(n-1)/2}$, when $n$ is an odd integer

$(1+x)(1+x^2{)}^{-1}$
$=(1+x)[1-x^2+x^4-x^6...\infty ]$
$=[1-x^2+x^4-x^6...\infty ]+x[1-x^2+x^4-x^6...\infty ]$
$=[1-x^2+x^4-x^6...\infty ]+[x-x^3+x^5-x^7...\infty ]$
Hence both A and D are the correct answers.