Question

1.求 S_n=1/2+3/(2^2)++(2n-1)/(2^n

Answer 1

问：1.求 S_n=1/2+3/(2^2)++(2n-1)/(2^n

答：极限存在的前提下： 令 $S = \frac{1}{2} + \frac{3}{2^2} + \ldots + \frac{2n-1}{2^n}$ $2S = 1 + \frac{3}{2} + \ldots + \frac{2n-1}{2^{n-1}}$ $S = 2S - S = 1 + 1 + \frac{1}{2} + \ldots + \frac{1}{2^{n-2}} - \frac{2n-1}{2^n}$ $= 1 + \frac{1 - \frac{1}{2^{n-1}}}{1 - \frac{1}{2}} - \frac{2n-1}{2^n}$ $= 3 - \frac{1}{2^{n-2}} - \frac{2n-1}{2^n}$ $\lim_{\text{n} \to \infty} \left( \frac{1}{2} + \frac{3}{2^2} + \ldots + \frac{2n-1}{2^n} \right) = \lim_{\text{n} \to \infty} (3 - \frac{1}{2^{n-2}} - \frac{2n-1}{2^n}) = 3.$