Answer to Question #313200 in Real Analysis for Jyo

Question

Prove the limit of 𝑥𝑛 =1/2 [𝑥𝑛−1 + 𝑥𝑛]

Answer 1

$xn=\frac{1}{2}(Xn-1 +xn)$

$\lim xn=x1+lim \sum_{k=1}^n(x_{k+1}-x_k)$

$=x_1+x_2-x_1+lim \sum_{k=2}^n\frac{(-1)^{k-1}c}{2^{k-1}}$

$$=x_2+ lim \sum_{k=1}^{\frac{n}{2}}\frac{c}{2^{2k}}-lim \sum_{k=1}^{\frac{n}{2}}\frac{c}{2^{2k-1}}$$

$=x_2+c lim(\sum_{k=1}^{\frac{n}{2}}\frac{1}{4^k}-\sum_{k=1}^{\frac{n}{2}}\frac{1}{2}\frac{1}{2^{2k-1}})$

$=x_2+c(\frac{1}{4}- (\sum_{k=0}^{\frac{n-1}{2}}\frac{1}{2}\frac{1}{4^k})$

$$=x_2+c(\frac{1}{4}-\frac{2}{3}) =x_2+\frac{2}{3}c$$