Convergence of \sum_{n=1}^\infty \frac{1}{n!}: A Basic Comparison Test

In summary, the first conversation is discussing how to compare the infinite sum \sum_{n=1}^\infty \frac{1}{n!} to a p-series and the second conversation is discussing how to compare the infinite sum \sum_{n=1}^\infty \frac{2^n}{n^2} to something else, such as 2^n or the harmonic series. The limit test is mentioned as a method to prove convergence or divergence, but the second person is unsure if it is allowed in the problem.
  • #1
PCSL
146
0
[tex]\sum_{n=1}^\infty \frac{1}{n!}[/tex]

I understand what n! means, but I have no clue what to compare this to. It is obvious to me that the sum converges, but I'm not sure how to prove it. I assume I would compare it to a p-series but I need help. Thanks!
 
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  • #2
Compare it to [itex]1/n^2[/itex]...
 
  • #3
micromass said:
Compare it to [itex]1/n^2[/itex]...

lol, I just realized how simple this is. my bad.
 
  • #4
Sorry, one more.

[tex]\sum_{n=1}^\infty \frac{2^n}{n^2}[/tex]

What would I compare this to?

I can clearly see that it diverges since numerator is waaaay bigger but I don't know how to prove it.
 
  • #5
Calculate the limit of the terms and show that the limit isn't 0.
 
  • #6
micromass said:
Calculate the limit of the terms and show that the limit isn't 0.

Sorry I didn't specify. I understand how to use the limit test. For this problem I am supposed to compare it to something. Thanks for putting up with my questions :).

edit: since 2^n is soo much bigger then n^2 can I compare it to 2^n?
 
  • #7
Maybe the harmonic series??

It's a stupid exercise anyway if you're not allowed to do the limit test.
 

1. What is the Basic Comparison Test?

The Basic Comparison Test, also known as the Limit Comparison Test, is a method used to determine the convergence or divergence of an infinite series. It compares the given series to a known series with known convergence or divergence.

2. How do you perform the Basic Comparison Test?

To perform the Basic Comparison Test, you first need to find a known series that has the same general behavior as the given series. Then, you take the limit of the ratio of the given series to the known series. If the resulting limit is a finite positive number, then both series have the same convergence or divergence. If the limit is 0, then the given series converges, and if the limit is infinity, then the given series diverges.

3. What is the difference between the Basic Comparison Test and the Direct Comparison Test?

The Basic Comparison Test and the Direct Comparison Test are similar in that they both involve comparing a given series to a known series. However, the Basic Comparison Test uses the limit of the ratio of the two series, while the Direct Comparison Test compares the terms of the two series directly.

4. Can the Basic Comparison Test be used for all types of series?

No, the Basic Comparison Test can only be used for series with positive terms. It cannot be used for series with alternating terms, such as the Alternating Series Test, or for series with negative terms.

5. What is the significance of using the Basic Comparison Test?

The Basic Comparison Test is a useful tool for determining the convergence or divergence of a series. It can be used to simplify complex series and make them easier to analyze. It can also help to determine which series are similar in behavior and which are significantly different.

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