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.

  • $\begingroup$ @rhombicosicodecahedron Those are nonnegative sequences and the question specifically asks for nonpositive sequences $\endgroup$ – 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.