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We know that, if we have two series with positive terms, the following holds:

If $\lim _{n\to \infty }\left(\frac{a_n}{b_n}\right)=1$ then the series $\sum _{n=1}^{\infty }\:a_n$ and $\sum _{n=1}^{\infty }\:b_n$ are of the same nature(the first series converges if and only if the second one coverges or the first one diverges if and only if the second one diverges)

This property is not true for series that can have terms of different signs and I'm trying to find and example of two series whose terms do fulfill the condition $\lim _{n\to \infty }\left(\frac{a_n}{b_n}\right)=1$ but are not of the same nature.

Can anyone help me with it?

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    @YuriyS Not actually. They can also be irrational numbers. The condition is for the terms not to be complex numbers. – 2017-02-06

1 Answers 1

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$$a_n=\frac{(-1)^n}{\sqrt{n}}$$ $$b_n=\frac{(-1)^n}{(-1)^{n}+\sqrt{n}}$$

Then:

$$\lim \frac{a_n}{b_n} = 1$$

(1) $\sum a_n$ converges but (2) $\sum b_n$ diverges.

(1) Alternating series tests

(2) Write:

$$b_n=\frac{(-1)^n}{\sqrt{n}}\frac{1}{1+\frac{(-1)^{n}}{\sqrt{n}}}=\frac{(-1)^n}{\sqrt{n}}\left(1-\frac{(-1)^n}{\sqrt{n}}+o\left(\frac{1}{n}\right)\right)$$

Thus:

$$b_n=\frac{(-1)^n}{\sqrt{n}}-\frac{1}{n}+o\left(\frac{1}{n\sqrt{n}}\right)$$

$\sum \frac{1}{n}$ diverges and the two other converge, so $\sum b_n$ diverges.

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    $\sum b_n$ converges by Leibniz's criterion (alternating series whose general term goes to $0$). – 2017-02-06
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    @Fimpellizieri: This criterion requires also $|b_n|$ to be decreasing, which is not the case. – 2017-02-06
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    Oops, you are correct. – 2017-02-06