Question

Пределы с прогрессией

Answer 1

1)
$lim_{n to infty} ( frac{1}{n^2} +frac{2}{n^2} +...+frac{n-1}{n^2} )= lim_{n to infty} (frac{1+2+...+(n-1)}{n^2} )$$= lim_{n to infty} frac{n(n-1)}{2n^2} = frac{1}{2}$

2)
$lim_{n to infty} (3^{ frac{1}{2}}+ 3^{-frac{1}{2}}+3^{- frac{3}{2}}+...+3^{ frac{3}{2}-n } = lim_{n to infty} ( frac{sqrt{3} (3^n-1)}{ frac{2}{3}*3^n } ) = frac{3 sqrt{3} }{2}$

3)
$lim_{n to infty} ( frac{2}{10} + frac{4}{100}+...+ frac{2^n}{10^n} + frac{5}{10}+ frac{25}{100}+...+ frac{5^n}{10^n} )$$= lim_{n to infty} ( frac{2^n-1}{2^n}+ frac{5^n-1}{4*5^n} )=1+ frac{1}{4} = frac{5}{4}$