HallsofIvy said:A Taylor's series about what central point? Rather than work out a large number of derivatives, I would calculate a few derivatives at the given point and try to find a pattern.
For example, the Taylor's series about x= 0 (the McLaurin series), has f(0)= 1, f'(0)= 0, f''(0)= -2, f'''(0)= 0, f''''(0)= 4!, etc. so I would hypothesize that the nth derivative, at 0, is 0 for odd n, [itex](-1)^n n![/itex] for n odd. Then I would try to prove that is true by induction.
The Taylor series of 1/(1 + x^2) is an infinite series expansion that represents the function 1/(1 + x^2) as a sum of terms involving powers of x. It is commonly known as the Maclaurin series since it is centered at x = 0.
The general formula for the Taylor series of 1/(1 + x^2) is f(x) = 1 + x^2 + x^4 + x^6 + ... = ∑(n=0 to ∞) x^(2n). This means that each term in the series is given by x raised to an even power, starting with x^0 = 1 and increasing by 2 for each subsequent term.
The Taylor series of 1/(1 + x^2) is useful in many areas of mathematics, including calculus, complex analysis, and differential equations. It allows for approximating the function 1/(1 + x^2) with a polynomial, which can make it easier to solve problems involving this function.
The convergence interval of the Taylor series of 1/(1 + x^2) is -1 < x < 1. This means that the series will converge to the function 1/(1 + x^2) for any value of x within this interval. Outside of this interval, the series may not converge or may converge to a different function.
The Taylor series of 1/(1 + x^2) can be used to approximate the value of the function at a given point by plugging in the value of x into the series. The more terms that are included in the series, the more accurate the approximation will be. This method is commonly used in numerical analysis and can be particularly useful when the function is difficult to evaluate directly.