Does the Series $\sum_{n=1}^{\infty} \frac{n+2^n}{n+3^n}$ Converge or Diverge?

Thread starter Askhwhelp
Start date Oct 30, 2013
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Convergence Divergence

In summary, convergence and divergence refer to the behavior of a mathematical or scientific series. Convergence occurs when a series approaches a finite limit as the number of terms increases, while divergence occurs when the series does not approach a finite limit and instead diverges to infinity. There are various methods for determining convergence or divergence, such as the comparison test, the ratio test, and the integral test. These tests involve examining the behavior of the terms in the series and comparing them to known convergent or divergent series. Convergence and divergence can be observed in various real-life phenomena, such as population growth, economic trends, and weather patterns. A series can only either converge or diverge, and a series that exhibits both behaviors is considered to be diver

Oct 30, 2013

Askhwhelp

The sum is $$\sum_{n=1}^{\infty} \frac{n+2^n}{n+3^n}$$ Is this convergent or divergent? I tried to use the divergent test but the test fail because $a_n = (n+2^n)/(n+3^n) = 0 $ as $n$ goes to infinity. Could someone point me to the right direction?

Thanks

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Oct 30, 2013

Office_Shredder

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Science Advisor

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What are some other tests that you know? Can you apply any of them?

FAQ: Does the Series $\sum_{n=1}^{\infty} \frac{n+2^n}{n+3^n}$ Converge or Diverge?

1. What is the difference between convergence and divergence?

Convergence and divergence refer to the behavior of a mathematical or scientific series. Convergence occurs when a series approaches a finite limit as the number of terms increases, while divergence occurs when the series does not approach a finite limit and instead diverges to infinity.

2. How do you determine if a series converges or diverges?

There are various methods for determining convergence or divergence, such as the comparison test, the ratio test, and the integral test. These tests involve examining the behavior of the terms in the series and comparing them to known convergent or divergent series.

3. What are some real-life examples of convergence and divergence?

Convergence and divergence can be observed in various phenomena, such as population growth, economic trends, and weather patterns. For example, a population can converge to a stable size as resources become limited, while a hurricane can diverge and intensify as it gains energy and moves towards land.

4. Can a series both converge and diverge?

No, a series can only either converge or diverge. A series that exhibits both behaviors is considered to be divergent.

5. Why is understanding convergence and divergence important in science?

Convergence and divergence play a crucial role in mathematical and scientific analysis, as they help determine the behavior and predict the outcomes of various processes. This understanding is essential in fields such as physics, economics, and engineering, where precise calculations and predictions are necessary.