lim-x-1-1-x-1-2-x-2-2-1-3-x-1-3-1-x-x-1-x-

Question Number 179950 by mathlove last updated on 04/Nov/22

\lim_{x \to \infty} [(1 + \frac{1}{x}) + {(1 + \frac{2}{x})}^{\frac{2}{2}} + {(1 + \frac{3}{x})}^{\frac{1}{3}} + \cdot \cdot \cdot \cdot + {(1 + \frac{x}{x})}^{\frac{1}{x}}] = ?

Commented by mahdipoor last updated on 04/Nov/22

x > n > 0 \Rightarrow 1 + \frac{n}{x} > 1 \Rightarrow {(1 + \frac{n}{x})}^{1 / n} > 1^{1 / n}

\Rightarrow A (x) = \underset{n = 1}{\sum^{x}} {(1 + \frac{n}{x})}^{1 / n} > [\underset{n = 1}{\sum^{x}} (1^{1 / n} = 1)] = x

\Rightarrow \lim_{x \to \infty} (A (x)) = \infty