MHB Does the Series \(\sum_{n=1}^{\infty} \frac{5+n \cos n}{n^2+2^n}\) Converge?

[maxkor]

\sum_{n=1}^{ \infty } \frac{5+n cosn}{n^2+2^n} is convergent?

 

[Opalg]

\sum_{n=1}^{ \infty } \frac{5+n \cos n}{n^2+2^n} is convergent?

Have you tried using the ratio test?

 

[maxkor]

If $\left \{ a_n \right \}, \left \{ b_n \right \} > 0$, and the limit $ \lim_{n \to \infty} \frac{a_n}{b_n}$ exists, is finite and is not zero, then $\sum_{n=1}^\infty a_n$ converges if and only if $\sum_{n=1}^\infty b_n converges.$
But $\left \{ b_n \right \} =??$

 

[Euge]

\sum_{n=1}^{ \infty } \frac{5+n cosn}{n^2+2^n} is convergent?

Hi maxkor,

While I agree with the suggestion posted by Opalg, I think it should be used indirectly. More precisely, since

$$\left|\frac{5 + n\cos n}{n^2 + 2^n}\right| \le \frac{5 + n}{n^2 + 2^n} < \frac{5 + n}{2^n},$$

use the ratio test to show that $\sum\limits_{n = 1}^\infty \frac{5 + n}{2^n}$ converges, then conclude by comparison that $\sum\limits_{n = 1}^\infty \frac{5 + n\cos n}{2^n}$ converges.