Suppose that $a_n, b_n > 0$ for all $n$. Prove that if $0 < \lim(a_n/b_n) < \infty$, then $\sum a_n$ and $\sum b_n$ either both converge or both diverge.

I'm a bit unsure on how to proceed with the proof... Any suggestions? My class uses Ross' book.

  • $\begingroup$ Do you know something about Cauchy's principle for convergence of a series? You can use that criteria here. You only need to show that if $\sum a_{n}$ converges then $\sum b_{n}$ also converges. The reverse (about divergence) can be proved by exchanging $a_{n}, b_{n}$ and using contradiction. $\endgroup$ – Paramanand Singh Mar 9 '14 at 5:49

Let $L:=\lim_{n\to +\infty}\frac{a_n}{b_n}$ (which exists by assumption and is a positive real number). Using the definition of a limit with $\varepsilon =L/2$, we obtain a positive integer $N$ such that if $n\geqslant N$, then $$\frac L2\leqslant \frac{a_n}{b_n}\leqslant \frac{3L}2,$$ and since $b_n$ is positive, we obtain for $n\geqslant N$, $$\frac L2b_n\leqslant a_n\leqslant \frac{3L}2b_n.$$ We conclude using the following two facts:

  1. If $K\in\mathbb R$ and the series $\sum_{n\geqslant 0}c_n$ is convergent, so is the series $\sum_{n\geqslant 0}(Kc_n)$.
  2. If $(c_n)_n$, $(c'_n)_n$ are sequence of positive real numbers, the series $\sum_n c_n$ is convergent and $c'_n\leqslant c_n$ for each $n$, then the series $\sum_n c'_n $ is convergent.

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