MathDB

Let

1<n\in\mathbb{N}

and

1\le a\in\mathbb{R}

and there are

n

number of

x_i, i\in\mathbb{N}, 1\le i\le n

such that

x_1=1

and

\frac{x_{i}}{x_{i-1}}=a+\alpha _ i

for

2\le i\le n

, where

\alpha _i\le \frac{1}{i(i+1)}

. Prove that

\sqrt[n-1]{x_n}< a+\frac{1}{n-1}

19