Figuring out whether a series converges or not

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In summary, the conversation discusses using the ratio test to determine whether a series converges or diverges, as well as the conditions for convergence or divergence based on the value of L in the ratio test. The attempt at a solution attached also concludes that the series converges based on the result of the ratio test.
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eventob
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Homework Statement


I am supposed to show whether the series converges or diverges. I guess I need to use the ratio test … at least my calculus professor told us that we should use that test whenever we had complicated terms like n! and such.


Homework Equations


lim_n->inf a_(n+1)/a_n = L
If L<1 the series converges, if L>1 it diverges and if L=0 ... who knows? :p


The Attempt at a Solution


I tried to do the problem, but I am not sure if I got it right or not. My attempt at a solution is attached, along with the problem, in the image.


Thanks in advance.
 

Attachments

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  • #2
Sure. The ratio test gives you 0. It converges.
 
  • #3
eventob said:

Homework Equations


lim_n->inf a_(n+1)/a_n = L
If L<1 the series converges, if L>1 it diverges and if L=0 ... who knows? :p
0 < 1, so if L = 0 in the Ratio Test, the series converges.
 

1. What is the definition of convergence for a series?

The definition of convergence for a series is when the sum of all the terms in the series approaches a finite value as the number of terms increases towards infinity.

2. How can I tell if a series is convergent or divergent?

There are several tests that can be used to determine if a series is convergent or divergent, such as the limit comparison test, ratio test, and integral test. These tests involve evaluating the behavior of the terms in the series as the number of terms increases towards infinity.

3. What is the difference between absolute and conditional convergence?

Absolute convergence occurs when a series converges regardless of the order of the terms, while conditional convergence occurs when a series only converges if the terms are arranged in a specific order. For example, the alternating harmonic series is conditionally convergent but not absolutely convergent.

4. Can a series be both convergent and divergent?

No, a series cannot be both convergent and divergent. A series can either converge to a finite value or diverge to infinity or negative infinity. It cannot do both simultaneously.

5. How can I use the convergence or divergence of a series to solve real-world problems?

Convergence and divergence of series can be used in many real-world applications, such as in economics, physics, and engineering. For example, in economics, the concept of convergence is used to analyze the behavior of financial markets. In physics, convergence and divergence of series are used to model the behavior of electric circuits. In engineering, the convergence or divergence of series is used to determine the stability of a system.

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