Question

The value of $(1+ix+i^2{x}^2+i^3{x}^3+\dots to\infty )\times$
$(1-ix+i^2{x}^2-i^3{x}^3+\dots .to\infty )$ is

• A. Imaginary
• B. a positive real
• C. a negative real
• D. equal to zero

Answer 1

The correct option is B a positive real

We have
$A=(1+ix+i^2{x}^2+i^3{x}^3--)=(1+ix-x^2-{ix}^3+x^4+{ix}^5----)$
$=\frac{(1-x^2+x^4-x^6--)}{p}+i\frac{(x-x^3+x^5---)}{Q}$
Similarly
,$B=(1-ix+i^2{x}^2-i^3{x}^3--)=(1-ix-x^2+x^3+x^4-i{x}^5----)$
$=\frac{(1-x^2+x^4--)}{P}-i\frac{(x-x^3+x^5--)}{Q}$
So $A\ast B=(P+iQ)\ast (P-iQ)$
$P^2+Q^2=$real (positive)