Series, Comparison and Limit Tests

In summary, the author is practicing problems for a Calc 2 test and found that the limit comparison test yielded the correct answer. This information is irrelevant, as the Comparison Test would have to be passed in order for the series to diverge.
  • #1
mateomy
307
0
Im practicing problems for my Calc 2 test tomorrow. I am doing this problem which I am not quite sure I've done it right. I think I have, but I want confirmation...

[tex]
\sum_{n=1}^{\infty} \frac{1}{\sqrt(n+4)}
[/tex]

I chose my comparison series as [itex]\frac{1}{\sqrt(n)}[/itex] and then ran a limit test eventually finding that it came out to equal 1, which shows that they are behaviorally similar. And since I know my comparison Series is larger than the given Series, I know the latter diverges.

Is that correct?
 
Physics news on Phys.org
  • #2
mateomy said:
Im practicing problems for my Calc 2 test tomorrow. I am doing this problem which I am not quite sure I've done it right. I think I have, but I want confirmation...

[tex]
\sum_{n=1}^{\infty} \frac{1}{\sqrt(n+4)}
[/tex]

I chose my comparison series as [itex]\frac{1}{\sqrt(n)}[/itex] and then ran a limit test eventually finding that it came out to equal 1, which shows that they are behaviorally similar. And since I know my comparison Series is larger than the given Series, I know the latter diverges.

Is that correct?

You can omit the phrase "And since I know my comparison Series is larger than the given Series".

Since the ratio in the Limit Comparison Test (which is what you used) is 1, and since your comparison series is a divergent series, then your series diverges as well.

The fact that your series is larger than or smaller than (on a per-term basis) the corresponding term of a divergent series, is irrelevant. If you were using the Comparison Test, however, this information would be relevant. In the Comparison Test, for your series to diverge, each term of your series would have to be larger than the corresponding term of a divergent series.
 
  • #3
I know, I realized that after I posted it. Is it correct otherwise?
 
  • #4
Thanks. (I had to delete a follow up question because I just jumped on a response after skimming over yours.)
 

1. What is a series test?

A series test is a mathematical tool used to determine the convergence or divergence of a series. It involves comparing the series to a known series that either converges or diverges, and using that information to make a conclusion about the original series.

2. What is a comparison test?

A comparison test is a method used to determine the convergence or divergence of a series by comparing it to a known series with similar properties. It involves comparing the terms of the original series to the terms of the known series, and using that information to make a conclusion about the convergence or divergence of the original series.

3. What is a limit test?

A limit test is a method used to determine the convergence or divergence of a series by taking the limit of its terms. If the limit is equal to 0, the series may converge, while if the limit is not equal to 0, the series must diverge.

4. How do I know which series test to use?

The specific series test to use depends on the properties and behavior of the series. For example, if the series contains only positive terms, the comparison test or limit test may be appropriate. If the series contains alternating signs, the alternating series test may be used. It is important to carefully analyze the series and choose the appropriate test based on its characteristics.

5. What is the difference between convergence and divergence of a series?

A series is said to converge if the sum of its terms approaches a finite value as the number of terms increases. On the other hand, a series is said to diverge if the sum of its terms increases without bound as the number of terms increases. In other words, a convergent series has a finite sum, while a divergent series does not have a finite sum.

Similar threads

  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
55
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
29
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
640
  • Calculus and Beyond Homework Help
Replies
1
Views
484
  • Calculus and Beyond Homework Help
Replies
17
Views
1K
  • Calculus and Beyond Homework Help
Replies
14
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
645
  • Calculus and Beyond Homework Help
Replies
7
Views
945
Back
Top