# Application of Comparison Test

CONTEXT: Uni question made up by lecturer.

If $$b_n \le a_n \le 0$$ and $$\sum b_n$$ is convergent, does this mean $$\sum a_n$$ is also convergent? I'm pretty sure this is true but can't work out how to apply the comparison test to it.

The comparison test states:

(1) If $$\sum_{n=0}^\infty b_n$$ is convergent and $$0 \le a_n \le b_n$$, then $$\sum_{n=0}^\infty a_n$$ is also convergent.

(2) If $$\sum_{n=0}^\infty b_n$$ is divergent and $$a_n \ge b_n \ge 0$$, then $$\sum_{n=0}^\infty a_n$$ is also divergent.

I can't apply (1) or (2) because $$b_n \le a_n \le 0$$ does not fit with either of these.

• @rhombicosicodecahedron Those are nonnegative sequences and the question specifically asks for nonpositive sequences – Tony S.F. May 6 at 8:33

Just apply the comparison test to $$-a_n$$ and $$-b_n$$.

Let $$\tilde{a_n} = -a_n$$ and $$\tilde{b_n}=-b_n$$.

We have,

$$0\leq \tilde{a_n} \leq \tilde{b_n}$$

Let $$B=\sum b_n$$ which is finite by assumption.

$$\sum \tilde{b_n} = \sum -b_n = -\sum b_n = -B$$

so $$\tilde{b_n}$$ is convergent. Apply the comparison test to see that $$\tilde{a_n}$$ is convergent.