Thanks for catching that! I will make the corrections.

In summary, we are trying to find the limit of $\displaystyle \left[\frac{\arctan{(n)}}{\pi +\arctan{(n)}}\right]$ as $n$ approaches infinity, which is equal to $\frac{1}{3}$. L'Hopital's rule does not apply here, so the limit can be evaluated by considering the behavior of the arctangent function as $n$ approaches infinity. This function approaches $\frac{\pi}{2}$, which can be simplified to $\frac{1}{3}$. It is important to note that when using the variable $n$, it is a sequence, while using $x$ implies a function.
  • #1
karush
Gold Member
MHB
3,269
5
$\tiny{242.ws8.d}$
$$\displaystyle
L_d=\lim_{x \to \infty}
\left[\frac{\arctan{(n)}}{\pi +\arctan{(n)}}\right]
=\frac{1}{3}$$

$\text{L' didn't work}$

☕
 
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  • #2
That's not an indeterminate form, so L'Hopital's rule does not apply. The limit may be found by evaluating the numerator and denominator separately. Do you see how?
 
  • #3
karush said:
$\tiny{242.ws8.d}$
$$\displaystyle
L_d=\lim_{x \to \infty}
\left[\frac{\arctan{(n)}}{\pi +\arctan{(n)}}\right]
=\frac{1}{3}$$

$\text{L' didn't work}$

☕

First of all, this is not a series, as you aren't summing up terms. This is a sequence. Anyway

$\displaystyle \begin{align*} \frac{\arctan{(n)}}{\pi + \arctan{(n)}} &= \frac{1}{\frac{\pi}{\arctan{(n)}} + 1} \end{align*}$

What happens to the arctangent function as $\displaystyle \begin{align*} n \to \infty \end{align*}$?
 
  • #4
Prove It said:
This is a sequence.

That's odd - it looks like a limit to me - possibly part of a convergence test for a series that is not given. Why do you think it's a sequence?
 
  • #5
greg1313 said:
That's odd - it looks like a limit to me - possibly part of a convergence test for a series that is not given. Why do you think it's a sequence?

The convention is that when "n" is used as the variable, it's a sequence, while where "x" is used is a function.
 
  • #6
Prove It said:
$\displaystyle \begin{align*} \frac{\arctan{(n)}}{\pi + \arctan{(n)}} &= \frac{1}{\frac{\pi}{\arctan{(n)}} + 1} \end{align*}$

What happens to the arctangent function as $\displaystyle \begin{align*} n \to \infty \end{align*}$?

$$\arctan(\infty)\implies\frac{\pi}{2}$$

$$\frac{1}{\frac{\pi}{\pi/2} + 1}
=\frac{1}{2+1}=\frac{1}{3}$$
 
  • #7
karush said:
$$\arctan(\infty)\implies\frac{\pi}{2}$$

Actually, I think you'd want to write

$$\lim_{n\to\infty}\arctan(n)=\frac{\pi}{2}$$

Note that $\tan(u)$ is one to one over the interval $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ and that $\tan(u)\to\infty$ as $u\to\frac{\pi}{2}$.

So,

$$\lim_{n\to\infty}\arctan(n)=\lim_{u\to\pi/2}\arctan(\tan u)=\frac{\pi}{2}$$

Also, you're writing

$$\lim_{x\to\infty}$$

in several places where the limit is with respect to $n$, so you should be writing

$$\lim_{n\to\infty}$$
 

FAQ: Thanks for catching that! I will make the corrections.

1. What is the significance of "242.ws8.d" in the series limit?

The term "242.ws8.d" is likely a specific identifier or label used in a particular scientific study or experiment. It may refer to a sample, measurement, or other data point within the series. The exact meaning can only be determined by referencing the specific context in which it was used.

2. How is the limit of a series determined?

The limit of a series is determined by analyzing the behavior of its terms as the number of terms increases towards infinity. This can involve various techniques such as the ratio test, comparison test, or integral test. The limit is the value that the series approaches as the number of terms increases.

3. Can the limit of a series be infinite?

Yes, the limit of a series can be infinite. This occurs when the terms of the series continue to increase or decrease without bound as the number of terms increases. In this case, the series does not converge to a specific value, and the limit is considered to be infinite.

4. What is the difference between a convergent and a divergent series?

A convergent series is one in which the terms approach a specific limit as the number of terms increases. In contrast, a divergent series is one in which the terms do not approach a specific limit and instead either increase or decrease without bound. Convergent series are considered to have a finite sum, while divergent series do not.

5. How is the limit of a series used in scientific research?

The limit of a series is a fundamental concept in mathematical analysis and is used in various fields of science, including physics, chemistry, and engineering. It is often used to model and predict the behavior of natural systems, such as population growth, chemical reactions, or physical processes. By understanding the limit of a series, scientists can make accurate predictions and draw conclusions about the behavior of these systems.

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