Prove-that-there-exists-M-gt-0-such-that-for-any-positive-integers-n-we-have-1-2-n-1-M-

Question Number 113641 by ZiYangLee last updated on 14/Sep/20

Prove that there exists M > 0 such that

for any positive integers n, we have

\sqrt{1 + \sqrt{2 + \sqrt{\dots + \sqrt{n + 1}}}} ⩽ M

Commented by mr W last updated on 14/Sep/20

A_{n} = \sqrt{1 + \sqrt{2 + \sqrt{3 + \dots \sqrt{n}}}}

A_{n} > \sqrt{1 + \sqrt{1 + \sqrt{1 + \dots \sqrt{1}}}} = C

1 + C = C^{2}

C^{2} - C - 1 = 0

C = \frac{1 + \sqrt{5}}{2}

A_{n} < \sqrt{n + \sqrt{n + \sqrt{n + \dots \sqrt{n}}}} = D

n + D = D^{2}

D^{2} - D - n = 0

D = \frac{1 + \sqrt{1 + 4 n}}{2}

\frac{1 + \sqrt{5}}{2} < A_{n} < \frac{1 + \sqrt{1 + 4 n}}{2}

w i t h M = ⌈ \frac{1 + \sqrt{1 + 4 n}}{2} ⌉ w h i c h a l w a y s e x i s t s

A_{n} < M

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