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I'm having difficulties with the following problem:

Let: $$ \begin{cases} x_n = n(1-a^{1\over n})\\ a > 0 \\ a \ne 1 \\ n \in \mathbb N \end{cases} $$ Show that $\{x_n\}$ is an increasing sequence.

I've already tried some default ways of proving such things with the help of division/subtraction but couldn't infer the right inequality from that. I've also played around with logarithms without any luck.

Could anyone give me a hint? (If it would be "divide" or "subtract" then please show some initial steps because i've already tried some of those).

Also please find the visualization via this link.

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  • $\begingroup$ So it should work in both $a>1$ and $0<a<1$ cases, is that right? Did you try some of each type numerically? [might be good to include such examples, several terms each] $\endgroup$ – coffeemath Nov 10 '18 at 13:46
  • $\begingroup$ @coffeemath not exactly numerically, but i've indeed been looking into a visualization of the sequence here $\endgroup$ – roman Nov 10 '18 at 13:56
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Use, the inequality, $b^m-1>m(b-1)$ where $m>1,b(\ne 1)>0$.

Now, take $b=a^{\frac{1}{n+1}},m=\frac{n+1}{n}$ and we are done using the above inequality.

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  • $\begingroup$ This worked like a charm, could you please point to the source of this inequality, I would like to see its proof? $\endgroup$ – roman Nov 12 '18 at 17:55

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