Evaluate Limit of Cosine Over Natural Logarithm

In summary: If you're going to assume anything, assume that you should work the problem in the simplest way you can get away with. By "get away with," I don't include any technique that contravenes explicit requirements in the problem.
  • #1
wilcofan3
27
0

Homework Statement



Evaluate [tex]\lim_{n->\infty} \frac {cos(n+1)} {ln n}[/tex]

Homework Equations



[tex]cos (x+1) = 1 - \frac {(x+1)^2} {2!} + \frac {(x+1)^4} {4!} - ...[/tex]

[tex]ln x = (x-1) - \frac {(x-1)^2} {2} + \frac {(x-1)^3} {3} - \frac {(x-1)^4} {4} + ...[/tex]

The Attempt at a Solution



Using those formulas, and then canceling, here is my pitiful attempt:

[tex]\lim_{n->\infty} \frac {1 - \frac {1} {n!}} {(n-1)}[/tex]
 
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  • #2
Can't you just take the limit directly? The numerator is always between -1 and 1, while the denominator grows large without bound.
 
  • #3
Mark44 said:
Can't you just take the limit directly? The numerator is always between -1 and 1, while the denominator grows large without bound.

I just didn't think I would simply be evaluating a limit in the section of Calculus that I'm in, so I assumed there was more to it involving series or something.

Any ideas?
 
  • #4
Does the problem say you have to use series representations? If not, the approach I gave is much simpler.
 
  • #5
Mark44 said:
Does the problem say you have to use series representations? If not, the approach I gave is much simpler.

The problem is from a worksheet that appears in the section of the book about series, but all the problem says is "Evaluate the limit." The other problems on the sheet are about series, and that's why I thought I had to do it this way.
 
  • #6
Limits often show up when you're asked to determine whether a given series converges or diverges, so maybe this limit will show up in a later problem on this sheet.
 
  • #7
Mark44 said:
Limits often show up when you're asked to determine whether a given series converges or diverges, so maybe this limit will show up in a later problem on this sheet.

Thanks.

So, just to make sure, since the numerator is between the interval -1<x<1 and the denominator goes off to [tex]\infty[/tex], this would mean the limit is going to 0, correct?

Is there anything I need to specifically show other than those aspects?
 
  • #8
Yes, correct.
You can use what some textbooks call the "Squeeze" or "Squeeze Play" theorem.

You can bound your limit expression like so:
-1/(ln n) <= cos(n + 1)/(ln n) <= 1/(ln n)

The limit, as n approaches infinity of the expression is 0, and the limit of the expression on the right is also 0, which means that the expression in the middle has the same limit.
 
  • #9
Thanks! Hopefully this is all that's required.
 
  • #10
wilcofan3 said:
I just didn't think I would simply be evaluating a limit in the section of Calculus that I'm in, so I assumed there was more to it involving series or something.

If you're going to assume anything, assume that you should work the problem in the simplest way you can get away with. By "get away with," I don't include any technique that contravenes explicit requirements in the problem.
 

1. What is the concept of evaluating a limit using series?

Evaluating a limit using series is a mathematical technique for finding the value of a limit by using a series, or a sum of infinitely many terms. This method is often used when the traditional methods of finding limits, such as direct substitution or L'Hopital's rule, do not work.

2. How do you determine if a limit can be evaluated using series?

There are a few criteria that can help determine if a limit can be evaluated using series. One of the main criteria is that the function must have a denominator that approaches infinity, or a numerator that approaches zero, as the variable in the limit approaches a specific value. Additionally, the function must also be continuous and differentiable in the interval of interest.

3. What are some common series used to evaluate limits?

Some common series used to evaluate limits include the geometric series, the harmonic series, and the Taylor series. These series have specific formulas that can be used to find their sum, making them useful for evaluating limits. Other series, such as the alternating series, can also be used in certain cases.

4. Can series always be used to evaluate limits?

No, series cannot always be used to evaluate limits. In some cases, the series may not converge, meaning that the sum of the terms does not approach a finite value. In these cases, the limit cannot be evaluated using series. Additionally, series may not be the most efficient method for finding a limit, so other techniques may be used instead.

5. How do we know if the limit found using series is accurate?

Series are often used to approximate values, rather than finding them exactly. However, the accuracy of the approximation can be improved by using more terms in the series. In general, the more terms used, the more accurate the approximation will be. In some cases, the exact value of the limit can be found by using an infinite number of terms in the series.

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