Question

Value of $\underset{n\to \infty }{lim}\frac{S_1{S}_n+S_2{S}_{n-1}+...+S_n{S}_1}{S_1^2+S_2^2+...+S_n^2}$ is

• A. $\frac{1}{8}$
• B. $\frac{1}{4}$
• C. $\frac{1}{2}$
• D. $1$

Answer 1

The correct option is C $\frac{1}{2}$

$\underset{n\to \infty }{lim}\frac{S_1{S}_n+S_2{S}_{n-1}+...+S_n{S}_1}{S_1^2+S_2^2+...+S_n^2}$
$=\underset{n\to \infty }{lim}\frac{\frac{n}{3}(2n^2+12n+13)}{\frac{n}{3}(4n^2+12n+11)}$
$=\frac{1}{2}$