Question

cho các số thực a,b thỏa |a|<1, |b|<1. tìm giới hạn I = lim (1+a+a^2+...+a^n)/(1+b+b^2+...+b^n) =

Answer 1

\(I=\lim\limits\dfrac{1+a+a^2+...+a^n}{1+b+b^2+...+b^n}\)

Xet tren tu la 1 csc voi : \(\left\{{}\begin{matrix}u_1=1\\q=a\end{matrix}\right.\Rightarrow S_a=1.\dfrac{a^{n+1}-1}{a-1}\)

Tuong tu cho mau so: \(S_b=1.\dfrac{b^{n+1}-1}{b-1}\)

\(\Rightarrow.....=\lim\limits\dfrac{\dfrac{a^{n+1}-1}{a-1}}{\dfrac{b^{n+1}-1}{b-1}}=\dfrac{\dfrac{1}{a-1}}{\dfrac{1}{b-1}}=\dfrac{1-b}{1-a}\)