Convergent Series?

In summary, the first series diverges while the second series converges. By bounding the internal sum in each series, it can be shown that the first series diverges due to its harmonic nature, while the second series converges due to its exponential nature.
  • #1
bomba923
763
0
(This isn't homework :redface:)

Does this series converge?
[tex]{\sum\limits_{n = 1}^\infty {\left[ {\left( {\sum\limits_{k = 1}^n {\frac{1}{k}} } \right)^{ - 1} } \right]} } [/tex]

Does this series converge?
[tex]{\sum\limits_{n = 1}^\infty {\left[ {\left( {\sum\limits_{k = 1}^n {k!} } \right)^{ - 1} } \right]} } [/tex]

**I would appreciate any help
 
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  • #2
The second one obviously converges, and almost as obviously the first diverges. (I hope...)

Bound the internal sum above by something and below by something in the first and second respectively, so that after inverting you're bounding each term in the big sum below and above. If done correctly you see that the first diverges (hint: think harmonic) and the second converges (think exponential)
 
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  • #3
you can give me.**

The first series, {\sum\limits_{n = 1}^\infty {\left[ {\left( {\sum\limits_{k = 1}^n {\frac{1}{k}} } \right)^{ - 1} } \right]} }, is known as the harmonic series and it is a well-known example of a divergent series. This means that the series does not have a finite sum and it continues to increase without bound as more terms are added.

On the other hand, the second series, {\sum\limits_{n = 1}^\infty {\left[ {\left( {\sum\limits_{k = 1}^n {k!} } \right)^{ - 1} } \right]} }, is known as the factorial series and it is also a divergent series. This series increases at a much faster rate than the harmonic series and it also does not have a finite sum.

In summary, neither of these series converge. They both continue to increase without bound and do not have a finite sum.
 

1. What is a convergent series?

A convergent series is a sequence of numbers that approaches a specific finite value as more terms are added. In other words, the sum of the terms in the series gets closer and closer to a certain number, known as the limit, as the number of terms increases.

2. How do you determine if a series is convergent?

There are several tests that can be used to determine if a series is convergent. Some of the most commonly used tests include the comparison test, the ratio test, and the integral test. These tests compare the given series to known convergent or divergent series, or use calculus to evaluate the limit of the series.

3. Can a series be both convergent and divergent?

No, a series can only be either convergent or divergent. If a series is convergent, it means that the sum of the terms approaches a finite value. If a series is divergent, it means that the sum of the terms approaches infinity or does not have a limit.

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

Absolute convergence refers to a series that converges regardless of the order of the terms. In other words, rearranging the terms of an absolutely convergent series will not change the sum. On the other hand, conditional convergence refers to a series that only converges when the terms are arranged in a specific order. Rearranging the terms of a conditionally convergent series can result in a different sum.

5. How are convergent series used in real-world applications?

Convergent series are used in many real-world applications, such as in finance, physics, and engineering. For example, in finance, the concept of compound interest can be represented as a convergent series. In physics, convergent series can be used to calculate the distance an object travels over time. In engineering, convergent series are used to model and predict the behavior of complex systems.

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