Is it possible to evaluate the summation of arctangents in this problem?

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In summary, the problem is to evaluate the limit of a summation involving arctangent, and the attempt involves using an identity to rewrite the summation and then applying a known procedure to evaluate it. The solution is obtained by using the identity and choosing a specific sequence, resulting in the limit being equal to pi/4.
  • #1
Saitama
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Problem:
Evaluate
$$\lim_{n\rightarrow \infty} \left(\sum_{r=1}^n (\arctan(2r^2))-\frac{n\pi}{2}\right)$$

Attempt:
I tried evaluating the summation but couldn't. Had the problem involved $\arctan(1/(2r^2))$, I could rewrite it as
$$\arctan\left(\frac{2r+1-(2r-1)}{1+(2r+1)(2r-1)}\right)$$
and evaluating the sum would be quite easy but honestly, I have no idea for the given problem.

Any help is appreciated. Thanks!
 
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  • #2
Pranav said:
I tried evaluating the summation but couldn't. Had the problem involved $\arctan(1/(2r^2))$, I could rewrite it as
$$\arctan\left(\frac{2r+1-(2r-1)}{1+(2r+1)(2r-1)}\right)$$
and evaluating the sum would be quite easy but honestly, I have no idea for the given problem.

Any help is appreciated. Thanks!

Maybe you could use that \(\displaystyle \arctan (x) + \arctan \left( \frac{1}{x}\right) =\frac{\pi}{2}\) for \(\displaystyle x>0\).
 
  • #3
Pranav said:
Problem:
Evaluate
$$\lim_{n\rightarrow \infty} \left(\sum_{r=1}^n (\arctan(2r^2))-\frac{n\pi}{2}\right)$$

Attempt:
I tried evaluating the summation but couldn't. Had the problem involved $\arctan(1/(2r^2))$, I could rewrite it as
$$\arctan\left(\frac{2r+1-(2r-1)}{1+(2r+1)(2r-1)}\right)$$
and evaluating the sum would be quite easy but honestly, I have no idea for the given problem.

Any help is appreciated. Thanks!

Using the identity...

$\displaystyle \sum_{k=1}^{n} \tan^{-1} (2\ k^{2}) - n\ \frac{\pi}{2} = - \sum_{k=1}^{n} \tan^{-1} \frac{1}{2\ k^{2}}\ (1)$

... the series becomes $\displaystyle - \sum_{k=1}^{\infty} \tan^{-1} \frac{1}{2\ k^{2}}$. Following the procedure described in...

http://mathhelpboards.com/math-notes-49/series-inverse-functions-8530.html

... we have $\displaystyle G(u,v) = \frac{u - v}{1 + u\ v}$ and choosing the sequence $c_{k}= \frac{k}{k+1}$ which is stricktly increasing and tends to 1, we obtain $\displaystyle G(c_{k},c_{k-1}) = \frac{1}{2\ k^{2}}$ and finally... $\displaystyle \sum_{k=1}^{\infty} \tan^{-1} \frac{1}{2\ k^{2}} = \tan^{-1} 1 - \tan^{-1} 0 = \frac{\pi}{4}\ (2)$ Kind regards $\chi$ $\sigma$
 
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  • #4
chisigma said:
Using the identity...

$\displaystyle \sum_{k=1}^{n} \tan^{-1} (2\ k^{2}) - n\ \frac{\pi}{2} = \sum_{k=1}^{n} \tan^{-1} \frac{1}{2\ k^{2}}\ (1)$
Ah, that was quite silly on my part, I was so much involved in evaluating the summation, I did not think of the above identity, thanks a lot chisigma! :)

BTW, you have a sign error. :p
... the series becomes $\displaystyle \sum_{k=1}^{\infty} \tan^{-1} \frac{1}{2\ k^{2}}$. Following the procedure described in...
That's the same series I mentioned in my attempt, it's easy to evaluate this. :)
 

FAQ: Is it possible to evaluate the summation of arctangents in this problem?

What is the summation of arctangents?

The summation of arctangents is a mathematical operation that involves adding together a series of arctangent values. It is commonly used in trigonometry and calculus to find the inverse tangent of a sum of angles.

How is the summation of arctangents calculated?

To calculate the summation of arctangents, you first need to convert each angle to radians. Then, you can use the formula: arctan(x) + arctan(y) = arctan((x + y) / (1 - xy)). This formula can be applied repeatedly to find the sum of more than two angles.

What are the properties of summation of arctangents?

The properties of summation of arctangents include: commutative property (changing the order of addition does not affect the result), associative property (grouping the angles differently does not affect the result), and distributive property (factoring out a constant does not affect the result).

What are some real-world applications of summation of arctangents?

Summation of arctangents is used in various fields such as engineering, physics, and astronomy. It can be used to calculate the total angle of rotation in a mechanical system, determine the trajectory of a projectile, and measure the curvature of space in astrophysics.

Are there any limitations to using summation of arctangents?

One limitation of summation of arctangents is that it only works for angles between -π/2 and π/2. Also, rounding errors can occur when performing multiple calculations, leading to inaccuracies in the final result. Additionally, the formula may not be applicable for complex or imaginary numbers.

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