Finding Power Series Representation for arctan(x): Help Needed!

In summary, to find the power series representation for arctan (x), you can start with the power series for 1/(1+x^2) and integrate term by term. Alternatively, you can differentiate the power series for 1/(1+x^2) and let x = 0 to get the power series for arctan (x). Hurkyl also gave a hint to try this approach.
  • #1
teng125
416
0
may i know how to solve this ques:find the power series representation for arctan (x)

i know that arctan (x) = integ 1/(1 + x^2) but then from here i don't know how to continue.
pls help...
 
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  • #2
It seems to me if you're looking for the power series of the thing on the left hand side, then you might want to try looking for the power series of the thing on the right hand side.
 
  • #3
ya I'm looking for the thing on the left hand side...pls help
 
  • #4
Take the thing on the right, and differentiate it a few times, and let x = 0 each time.

[tex]f(x) = f(0) + xf'(0) + \frac{x^2}{2!}f''(0) + ... [/tex]

Where f'(0) represents the derivative at x = 0. f''(0) is the second derivative etc. This will give a power series, then you can integrate term by term for the inverse tan function.
 
  • #5
One doesn't need to know any calculus at all to find the power series for [itex]1/(1+x^2)[/itex].


ya I'm looking for the thing on the left hand side...pls help
I gave you a big hint -- have you not tried to do anything with it? :grumpy:
 
  • #6
teng125 said:
ya I'm looking for the thing on the left hand side...pls help

Go back and read Hurkyl's reply again!
Hurkyl said:
It seems to me if you're looking for the power series of the thing on the left hand side, then you might want to try looking for the power series of the thing on the right hand side.
 

1. What is the power series representation for arctan(x)?

The power series representation for arctan(x) is n=0 ((-1)n/2n+1) x2n+1.

2. How is the power series for arctan(x) derived?

The power series for arctan(x) is derived by integrating the power series for 1/(1+x2), which is n=0 (-1)n x2n.

3. What is the interval of convergence for the power series of arctan(x)?

The interval of convergence for the power series of arctan(x) is -1 ≤ x ≤ 1.

4. Can the power series for arctan(x) be used to find an exact value for arctan(x)?

No, the power series for arctan(x) can only provide an approximation for arctan(x) within its interval of convergence. To find an exact value, other methods such as using a calculator or an inverse trigonometric function table should be used.

5. How can the power series for arctan(x) be used in real-world applications?

The power series for arctan(x) can be used in various mathematical and scientific calculations that involve the arctangent function, such as in solving differential equations and in numerical analysis. It can also be used in approximating the value of arctan(x) in situations where an exact value is not needed.

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