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How to Make a Power Series for an Inverse Tangent

A power series is a way to estimate the value a function of x for a particular value of x by using a series that includes powers of x.

The inverse tangent, or arctan, of x, is the function that yields the tangent when inverted. That is, if arctan(x) = y then tan(y) = x.

The tangent is a trigonometric function. In a right triangle, the tangent of an angle is the ratio of the side opposite the angle to the side adjacent to the angle.

There is a power series for arctan(x) when x is between -1 and 1.

Instructions

- 1
  Start with x. For example, suppose you want to find arctan(.5) using a power series. Start with .5.
- 2
  Find x^3. For the example, .5^3 = .125.
- 3
  Divide this by 3. In the example, this is .125/3 = .0417
- 4
  Subtract this from the previous result. In the example, .5 - .0417 = .4583.
- 5
  Find x^5 and divide by 5. For the example, this is .5^5/5 = .03125/4 = .00625.
- 6
  Add this to the previous result. In the example .4583 + .00625 = .46455.
- 7
  Add and subtract alternate terms until the desired accuracy is reached. The terms are of the form x^(2n-1)/(2n-1) for n starting at 1. Thus the first term (in step 1 above) was x^(2*1-1)/(2-1) = x^1/1 = x. The second term was x^(2*2-1)/(2*2-1) = x^3/3 (see step 3). The terms are alternately positive and negative, and the full series is
  
  x - x^3/3 + x^5/5 - x^7/7 ....

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