Given general term - find limits and comparison

In summary: What I wrote is the exact wording of the problem statement but I should have been more clear. The problem is from a test paper belonging to the section which consists of multiple choice questions. I have to select the correct options.
  • #1
Saitama
4,243
93

Homework Statement


Let ##\displaystyle a_n=\frac 1 2+\frac 1 3+...+\frac 1 n##. Then

A)##a_n## is less than ##\displaystyle \int_2^n\frac{dx}{x}##.

B)##a_n## is greater than ##\displaystyle \int_1^n\frac{dx}{x}##.

C)##\displaystyle \lim_{n\rightarrow \infty} \frac{a_n}{\ln n}=1##

D)##\displaystyle \lim_{n\rightarrow \infty} a_n## is finite.

Homework Equations


The Attempt at a Solution


To my knowledge, there is no known closed form for the given ##a_n##.

I am clueless about the right approach so I started with ##n=2##. For n=2, ##a_2=0.5##

Also,
$$\int_2^2 \frac{dx}{x}=0$$
and
$$\int_1^2 \frac{dx}{x}=\ln 2 \approx 0.693$$
Obviously, A and B are not the answers.

How do I check for other options?

Any help is appreciated. Thanks!
 
Physics news on Phys.org
  • #2
D is not correct the series is diverging. C is basically saying that a(n) = ln(n) if n becomes large is that correct?
 
  • #3
dirk_mec1 said:
D is not correct the series is diverging. C is basically saying that a(n) = ln(n) if n becomes large is that correct?

I don't have the answers at the moment, I will have them by tomorrow.

Can you please explain how do you get the series to be diverging?

Thanks!
 
  • #4
Think about comparing the series with approximating sums for the integrals in A and B.
 
  • #5
LCKurtz said:
Think about comparing the series with approximating sums for the integrals in A and B.

I am not sure if I understand your statement but do you ask me this:
$$a_n=\int_2^n \frac{dx}{x}$$
?
 
  • #6
LCKurtz said:
Think about comparing the series with approximating sums for the integrals in A and B.

Pranav-Arora said:
I am not sure if I understand your statement but do you ask me this:
$$a_n=\int_2^n \frac{dx}{x}$$
?

Yes. Think about approximating that with rectangles and see if you can relate it to your series.
 
  • Like
Likes 1 person
  • #7
LCKurtz said:
Yes. Think about approximating that with rectangles and see if you can relate it to your series.

Thanks LCKrutz! :)

I just found the wiki page on the given series titled "Harmonic Number" and it contains a nice sketch of approximation using rectangles. http://en.wikipedia.org/wiki/Harmonic_number

Getting back to the question, as ##n\rightarrow \infty##,
$$a_n=\ln n-\ln 2$$
##a_n## is obviously not finite. That leaves us with option C.

$$\lim_{n\rightarrow \infty} \frac{a_n}{\ln n}=\lim_{n\rightarrow \infty} \frac{\ln n-\ln 2}{\ln n}=\lim_{n\rightarrow \infty} 1-\frac{\ln 2}{\ln n}=1$$
Hence, C is correct, thanks a lot LCKurtz! :smile:
 
  • #8
What is the exact statement of the problem? Is it a True-False type question?

[Edit] I see you answered that question while I was posting it.
 
  • #9
LCKurtz said:
What is the exact statement of the problem? Is it a True-False type question?

What I wrote is the exact wording of the problem statement but I should have been more clear. The problem is from a test paper belonging to the section which consists of multiple choice questions. I have to select the correct options.
 

1. What is a general term in mathematics?

A general term in mathematics refers to a formula or expression that represents a sequence or pattern of numbers. It is usually denoted by the letter "n" and can be used to find specific terms in a sequence.

2. How do you find the limit of a general term?

To find the limit of a general term, you can use the formula lim(n→∞) an = L, where "an" is the general term and "L" is the limit. This formula represents the value that the general term approaches as "n" gets closer and closer to infinity.

3. What is the significance of finding limits in mathematics?

Finding limits is important in mathematics because it helps us understand the behavior of functions and sequences. It allows us to determine the value of a function at a certain point or to see how a sequence of numbers is approaching a specific value.

4. How can you compare two general terms?

Two general terms can be compared by finding their limits. If the limits of the two terms are equal, then they are considered equivalent. If one limit is greater than the other, then the corresponding general term will have a larger value for a given "n" value.

5. Are there any limitations to finding limits of general terms?

Yes, there are some limitations to finding limits of general terms. The general term must have a finite limit as "n" approaches infinity, and the limit must exist. Additionally, some more complex functions may require additional techniques, such as L'Hôpital's rule, to find the limit.

Similar threads

  • Calculus and Beyond Homework Help
Replies
13
Views
639
  • Calculus and Beyond Homework Help
Replies
8
Views
767
  • Calculus and Beyond Homework Help
Replies
34
Views
2K
  • Calculus and Beyond Homework Help
Replies
13
Views
403
  • Calculus and Beyond Homework Help
Replies
1
Views
90
  • Calculus and Beyond Homework Help
Replies
2
Views
660
  • Calculus and Beyond Homework Help
Replies
8
Views
588
  • Calculus and Beyond Homework Help
Replies
17
Views
487
  • Calculus and Beyond Homework Help
Replies
5
Views
436
  • Calculus and Beyond Homework Help
Replies
8
Views
938
Back
Top