Prove-that-1-1-2-1-3-1-n-lt-2-n-

Question Number 160694 by naka3546 last updated on 04/Dec/21

P r o v e t h a t

1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \dots + \frac{1}{\sqrt{n}} < 2 \sqrt{n}

Answered by mindispower last updated on 04/Dec/21

\frac{1}{\sqrt{1 + x}} = \frac{2}{2 \sqrt{1 + x}} < \frac{2}{\sqrt{x} + \sqrt{x + 1}} = 2 (\sqrt{1 + x} - \sqrt{x})

\underset{k = 0}{\sum^{n - 1}} \frac{1}{\sqrt{1 + k}} < 2 \underset{k = 0}{\sum^{n - 1}} (\sqrt{k + 1} - \sqrt{k})

1 + \frac{1}{\sqrt{2}} + \dots + \frac{1}{\sqrt{n}} < 2 \sqrt{n}

Commented by naka3546 last updated on 04/Dec/21

T h a n k y o u, s i r .

Commented by mindispower last updated on 05/Dec/21

w i t h e p l e a s u r g o d b l e s s Y o u