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

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In summary, we can use the ratio test to show that the series $\sum_{n=1}^{ \infty } \frac{5+n cosn}{n^2+2^n}$ is convergent, as it is bounded by the convergent series $\sum_{n=1}^\infty \frac{5 + n}{2^n}$. Therefore, by comparison, we can conclude that the original series is convergent.
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maxkor
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[tex]\sum_{n=1}^{ \infty } \frac{5+n cosn}{n^2+2^n}[/tex] is convergent?
 
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maxkor said:
[tex]\sum_{n=1}^{ \infty } \frac{5+n \cos n}{n^2+2^n}[/tex] is convergent?
Have you tried using the ratio test?
 
  • #3
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 $\left \{ b_n \right \} =??$
 
  • #4
maxkor said:
[tex]\sum_{n=1}^{ \infty } \frac{5+n cosn}{n^2+2^n}[/tex] 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.
 
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FAQ: Does the Series \(\sum_{n=1}^{\infty} \frac{5+n \cos n}{n^2+2^n}\) Converge?

What is a series problem?

A series problem is a mathematical problem that involves adding an infinite number of terms in a specific order. It is a type of mathematical series that follows a specific pattern and can be used to solve various problems in mathematics and science.

How do you determine if a series is convergent?

To determine if a series is convergent, you can use various tests such as the ratio test, comparison test, or integral test. These tests evaluate the behavior of the series and help determine if the sum of the terms approaches a finite value or diverges to infinity.

What does it mean for a series to be convergent?

A convergent series is a series in which the sum of the terms approaches a finite value as the number of terms increases. In other words, the series has a finite sum and does not diverge to infinity. This means that the series has a well-defined limit and can be evaluated accurately.

Can a series be both convergent and divergent?

No, a series cannot be both convergent and divergent. A series can either have a finite sum and be convergent or diverge to infinity and be divergent. It is not possible for a series to have both behaviors at the same time.

How is the convergence of a series related to its terms?

The convergence of a series is related to the behavior of its terms. If the terms of a series decrease in size, the series is more likely to be convergent. On the other hand, if the terms increase or oscillate, the series is more likely to be divergent. However, this is not always the case, and it is important to use convergence tests to determine the behavior of a series accurately.

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