Compute Arctangent Series: Sum from 0 to n

In summary, the conversation discusses trying to compute a sum involving arctangent and using trig identities to simplify it. The person is currently stuck on a specific part and asks if there is a formula for it. Two possible methods for solving the problem are also mentioned.
  • #1
amcavoy
665
0
I am trying to compute:

[tex]\sum_{k=0}^{n}\arctan{\left(\frac{1}{k^{2}+k+1}\right)}.[/tex]

I have used some trig identities and reduced it, but before I can do anything more I am stuck on:

[tex]\sum_{k=0}^{n}\arctan{\left(k\right)}.[/tex]

Is there a formula for this sum?

Thanks for the help.
 
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  • #2
I have two ways to do this (probably more, but two that I can think of):

[tex]\sum_{k=0}^{n}\arctan{\left(\frac{1}{k^{2}+k+1}\right)}=\frac{\pi n}{2}-\sum_{k=0}^{n}\arctan{\left(k^{2}+k+1\right)},[/tex]

and

[tex]\sum_{k=0}^{n}\arctan{\left(\frac{1}{k^{2}+k+1}\right)}=\sum_{k=0}^{n}\arctan{\left(k+1\right)}-\sum_{k=0}^{n}\arctan{\left(k\right)}.[/tex]

Which one is better to work with? Thanks for the help.
 

1. What is the purpose of computing the arctangent series?

The arctangent series is used to approximate the value of the arctangent function, which is a mathematical function that calculates the angle between the x-axis and a given point on a unit circle. It is also used in various mathematical and scientific calculations.

2. How do you compute the arctangent series?

The arctangent series can be computed by using the formula: arctan(x) = x - x^3/3 + x^5/5 - x^7/7 + ... + (-1)^n * x^(2n+1) / (2n+1). This formula involves an infinite sum, but it can be approximated by only using a certain number of terms from the series.

3. What is the significance of the 'n' in the arctangent series sum from 0 to n?

The 'n' in the arctangent series represents the number of terms being used to approximate the arctangent function. The larger the value of 'n', the more accurate the approximation will be.

4. Can the arctangent series be used for any value of x?

No, the arctangent series has a limited range of convergence, which means it can only be used for certain values of x. Specifically, it can only be used for values of x between -1 and 1. For values outside of this range, the series will not converge and will not provide an accurate approximation.

5. How does the accuracy of the arctangent series improve as more terms are added?

The accuracy of the arctangent series improves as more terms are added because each term in the series adds a smaller and smaller contribution to the overall sum. This means that as more terms are added, the approximation gets closer to the actual value of the arctangent function.

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