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Suppose that $(s_n)$ and $(t_n)$ are sequences of positive numbers such that $\lim \limits_{n \to \infty} \frac{s_n}{t_n}$=a and that $(s_n)$ diverges to infinity. What can you conclude?

This problem has me a little confused. Is it true that $(t_n)$ must also diverge to infinity to get a limit that approaches a finite value?

2 Answers 2

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You have $aT_n\sim S_n$, so there are to cases :

  • If $S_n$ diverges with $S_n\rightarrow +\infty$, then $aT_n\rightarrow +\infty$, so $T_n\rightarrow +\infty$.
  • If $S_n$ does not have a limit. If $aT_n$ has a limit in $\bar {\mathbb{R}}$ then $S_n$ has the same limit. So $T_n$ does not have a limit.

Finally $T_n$ also diverges.

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Yes, $t_n $ must diverge to infinity.

Suppose that some subsequence $t_{n_k}$ is bounded. Then there must exist some sub subsequence (which we will denote $t_{k_r}$ to avoid things getting really messy) which converges to some number $t$. Since subsequences of a convergent sequence converge to the same limit, the subsequence $\frac{s_{k_r}}{t_{k_r}}$ converges to $a$.

Thus $$\lim s_{k_r} = \lim t_{k_r} \frac{s_{k_r}}{t_{k_r}} = ta$$ a clear contradiction. Hence all sub sequences of $t_n$ are unbounded, so we have the result.