Direct comparison test for convergence

In summary, the conversation discusses comparing the series \sum_{n=0}^{\infty}\frac{1}{n!} to other series to determine if it converges or diverges. The ratio test is mentioned as a method for showing convergence, and it is determined that the series converges. The importance of sequences and series to a math major is also discussed, and it is suggested to compare the series to \frac{1}{2^n} or \frac{1}{n^2} to show convergence.
  • #1
RadiationX
256
0
I'm supposed to compare the series

[tex]\sum_{n=0}^{\infty}\frac{1}{n!}[/tex]

to some other series to see if the one above converges or diverges. I have no idea of what to compare it to.

I know by the ratio test that the above series converges, that is if I'm doing the ratio test correctly.

[tex]\lim_{n\rightarrow\infty}\frac{1}{(n+1)!}n!\\=\frac{1}{n(n+1)}(1*n)=\\\frac{1}{(n+1)}[/tex]

since this limit is zero which is less than one the series converges
 
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  • #2
I'm thinking about majoring in mathematics. How important are sequences and series to a math major? I hate them BTW.
 
  • #3
Compare it to the sum of 1/2^n. If you can show this converges, and show that every term in this series is greater than the corresponding term in the original series, that one must also converge.
 
  • #4
Or even, and simpler, just compare it to [tex]\frac{1}{n^2}[/tex]. As soon as n> 2,
[tex]\frac{1}{n^n}< \frac{1}{n^2}[/tex]. Now, what do you know about the convergence of [tex]\Sigma_1^\infty\frac{1}{n^2}[/tex]?
 

Related to Direct comparison test for convergence

What is the direct comparison test for convergence?

The direct comparison test is a method used in calculus to determine whether an infinite series converges or diverges. It involves comparing the terms of a given series to those of a known series, and using the convergence or divergence of the known series to determine the convergence or divergence of the given series.

How do you perform a direct comparison test?

To perform a direct comparison test, you must first identify a known series whose convergence or divergence is known. Then, you compare the terms of the given series to the terms of the known series. If the terms of the given series are smaller in value than the terms of the known series, and the known series converges, then the given series also converges. If the terms of the given series are larger in value than the terms of the known series, and the known series diverges, then the given series also diverges.

What is the direct comparison test used for?

The direct comparison test is used to determine whether an infinite series converges or diverges. It is commonly used in calculus and real analysis to evaluate the convergence of series and to determine the behavior of functions.

Can the direct comparison test be used for all series?

No, the direct comparison test can only be used for series with positive terms. It cannot be used for alternating series or series with negative terms. In addition, the terms of the series being compared must also be non-negative, otherwise the comparison is not meaningful.

What is the difference between the direct comparison test and the limit comparison test?

The direct comparison test and the limit comparison test are two different methods used to determine the convergence or divergence of infinite series. The direct comparison test compares the terms of a given series to those of a known series, while the limit comparison test compares the ratio of the terms of a given series to those of a known series. The limit comparison test is typically used when the terms of the given series are more complicated and cannot be easily compared to another series.

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