Where did the cosines come from?

Calculus Level 2

Let $\sum\limits_{n=0}^\infty a_n$ and $\sum\limits_{n=0}^\infty b_n$ be two convergent numerical series with positive terms.

True or false? :

The series $\sum\limits_{n=0}^\infty \left(a_n\cos(b_nx)+b_n\sin(a_nx) \right)$ is convergent in $\mathbb{R}$ .

These problem is from a test by Professor Maria do Ceú.

Insufficient information True False

1 solution

Paulo Guilherme Santos
Jun 14, 2015

We know that $\sum\limits_{n=0}^\infty a_n$ and $\sum\limits_{n=0}^\infty b_n$ are two convergent series with positive terms, therefore the series $\sum\limits_{n=0}^\infty \left| a_n \right|$ and the series $\sum\limits_{n=0}^\infty \left| b_n \right|$ are convergent, and the series $\sum\limits_{n=0}^\infty \left( \left| a_n \right|+\left| b_n \right| \right)$
is also convergent. Let's compare the series $\sum\limits_{n=0}^\infty \left( a_n\cos(b_nx)+b_n\sin(a_nx) \right)$ with the series $\sum\limits_{n=0}^\infty \left| a_n\cos(b_nx)+b_n\sin(a_nx) \right|$ . We have that $\left| a_n\cos(b_nx)+b_n\sin(a_nx) \right| \leq \left| a_n\cos(b_nx)\right|+\left|b_n\sin(a_nx) \right| \leq\left| a_n\right|+\left|b_n \right|$ therefore, by a comparison test, we have that $\sum\limits_{n=0}^\infty \left( a_n\cos(b_nx)+b_n\sin(a_nx) \right)$ converges in $\mathbb{R}$ .

The problem does not state that the series $\sum a_n$ and $\sum b_n$ are convergent; it merely states that they are "numerical series with positive terms". If they are supposed to be convergent, then the problem should say so.

Jon Haussmann - 6 years ago

You're completely right! I've already changed the problem. Thank you!

Paulo Guilherme Santos - 6 years ago

Where did the cosines come from?

These problem is from a test by Professor Maria do Ceú.

1 solution

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