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An example of a sequence $\left \{ a_n \right \}$ such that $\lim_{n\rightarrow \infty }a_n=\infty$, but $\lim_{n\rightarrow \infty }a_{n+k} -a_n=0$ for each fixed posvitive integer $k$.

since $2^n, \ln(n), \frac{e^{2n}}{n}$ are sequences with limit of this sequence is $\infty$ is any one satisfies the above property

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HINT: Let $a_n=H_n=\sum_{\ell=1}^n\frac1\ell$.

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The only one of your three sequence that satisfies $\lim_{n\rightarrow \infty }a_{n+k} -a_n=0 $ is $\ln(n)$.

This is true because $\ln(n+k)-\ln(n) =\ln(1+k/n) < k/n \to 0 $ (since $\ln(1+x) < x$).

For $2^n$, $2^{n+k}-2^n =2^n(2^k-1) \to \infty $ for any $k > 0$.

For $s_n=\frac{e^{2n}}{n} $, you can similarly show that $s_{n+k}-s_n \to \infty$ for any $k > 0$.

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We have $\sqrt n \to \infty,$ and for each fixed $k,$

$$\sqrt {n+k} - \sqrt {n} = \frac{k}{\sqrt {n+k} + \sqrt {n}} \to 0.$$

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