Question

求和：x/(1+x^2) + x^2/(1+x^2)^2 + ...+ x^n/(1+x^2)^n .

Answer 1

仔细观察,不难得出,此数列为等比数列,不妨设a1=x/(1+x^2) ,且q=x/(1+x^2)

∴Sn=a1(1-q^n)/(1-q)=x/(1+x^2) {1-[x/(1+x²)]^n}/{1-（x/（1+x²）]

整理得：Sn=x/(x²-x+1)×{1-[x/(1+x²)]^n}