Testing Convergence: Comparison & Limit Comparison Tests

In summary, the first series diverges using the p-series and comparison test, the second one converges because p > 1, and the third one (assuming it is -\frac{(-1)^n}{9n}) converges because it is an alternating series with terms going to 0.
  • #1
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Test each of the following series for convergence by either the Comparison Test or the Limit Comparison Test.

[tex]\sum_{n=1}^\infty \frac{2n^4}{n^5+7}[/tex] this diverges using the p-series and comparison test right? p <1


[tex]\sum_{n=1}^\infty \frac{2n^4}{n^9+7}[/tex] and this converges right? because p > 1



[tex]\sum_{n=1}^\infty \frac{-1^n}{9n}[/tex] i think this also diverges cause p <1. (not sure about this one)

can someone check these real quick and tell me if I am correct?
 
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  • #2
First one, P = 1, it diverges.

2 is correct.

Hint for the last one

[tex] \sum_{n=1}^{\infty} - \frac{1^n}{9n} [/tex]

Whats [itex] 1^n [/itex] for positive n? It should be easy after that.
 
  • #3
I suspect he meant [tex] \sum_{n=1}^{\infty} -\frac{(-1)^n}{9n} [/tex] for the last one. That converges because it is an alternating series with terms going to 0.
 
  • #4
I was going to make that suggestion, but since his post mentions the comparison test, I decided that the way he originally posted was likely the way that he meant it (since the comparison test can't be used for conditionally convergent series).
 

Related to Testing Convergence: Comparison & Limit Comparison Tests

1. What is the purpose of testing convergence in a series?

Testing convergence in a series is important because it helps determine whether a given series will converge or diverge. This is crucial in many areas of mathematics and science, as it allows us to make predictions and draw conclusions based on the behavior of a series.

2. What is the difference between the Comparison Test and the Limit Comparison Test?

The Comparison Test and the Limit Comparison Test are both used to determine the convergence or divergence of a series, but they differ in their approach. The Comparison Test compares the given series to a known convergent or divergent series, while the Limit Comparison Test compares the ratios of the terms in the given series to a known convergent series.

3. How do you determine which test to use when testing convergence?

The choice of which test to use when testing convergence depends on the given series and the information available about it. Generally, the Comparison Test is used when the series has similar terms or is a combination of known series, while the Limit Comparison Test is used when the terms in the series are complicated or do not have a clear pattern.

4. Can the Comparison and Limit Comparison Tests be used for all series?

No, the Comparison and Limit Comparison Tests cannot be used for all series. These tests have specific conditions that must be met in order for them to be applicable. It is important to carefully consider the properties of the series before applying these tests.

5. How can the Comparison and Limit Comparison Tests be used in real-world applications?

The Comparison and Limit Comparison Tests are commonly used in various fields of science and engineering to analyze and predict the behavior of mathematical models. For example, these tests can be used to determine the convergence of infinite series in physics and to analyze the performance of algorithms in computer science.

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