Exercise [04.05]

ZZR Puig · 24 May 2008, 17:42

We have:
${s}_{n}=1+x^2+x^4+x^6+x^8+x^{10}+x^{12}+...=\frac{1}{1-x^2}$
${s}_{m}=1-x^2+x^4-x^6+x^8-x^{10}+x^{12}-...=\frac{1}{1+x^2}$
Let's define s_k

as

so
$s_k=s_n+s_m=2+2x^4+2x^8+2x^{12}+...=\frac2{1-x^4}$
Then,
$s_m=s_k-s_n=\frac{2}{1-x^4}-\frac1{1-x^2}=\frac{2}{(1+x^2)(1-x^2)}-\frac1{1-x^2}=\frac{2-(1+x^2)}{(1+x^2)(1-x^2)}$
$s_m=\frac{1-x^2}{(1+x^2)(1-x^2)}=\frac1{1+x^2}$

P.S.: This is only a way to get the result without the use of complex numbers. The most elemental relation between both expressions, as stated by Sameed on the exercise discussion subforum, is the possibility to substitute x with ix to get the other one.