taylor series of 1 1 x 2

Guri Bee

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21-10-2024

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Unraveling the Taylor Series of 1/(1+x^2)

The Taylor series is a powerful tool in mathematics, allowing us to approximate functions using an infinite sum of terms. Today, we'll explore the Taylor series expansion of the function 1/(1+x^2). This function holds significance in calculus and is closely related to the arctangent function.

Understanding the Basics

Before diving into the series, let's recall the general form of a Taylor series expansion around a point a:

f(x) = f(a) + f'(a)(x-a)/1! + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ...

Here, f'(a), f''(a), etc. represent the first, second, and higher-order derivatives of the function f(x) evaluated at a.

The Taylor Series of 1/(1+x^2)

Let's focus on the Taylor series expansion of 1/(1+x^2) around the point a = 0. This is also known as the Maclaurin series. We need to find the derivatives of the function:

• f(x) = 1/(1+x^2)
• f'(x) = -2x/(1+x²⁾2
• f''(x) = (6x^2 - 2)/(1+x²⁾3
• f'''(x) = (-24x^3 + 24x)/(1+x²⁾4
• f''''(x) = (120x^4 - 120x^2 + 24)/(1+x²⁾5

Now, evaluating these derivatives at x = 0:

• f(0) = 1
• f'(0) = 0
• f''(0) = -2
• f'''(0) = 0
• f''''(0) = 24

Plugging these values into the Taylor series formula:

1/(1+x^2) = 1 + 0*x/1! - 2*x^2/2! + 0*x^3/3! + 24*x^4/4! + ...

Simplifying this, we get the Taylor series of 1/(1+x^2):

1/(1+x^2) = 1 - x^2 + x^4 - x^6 + ...

This series is an alternating series with a pattern of alternating signs and increasing powers of x.

Applications and Insights

The Taylor series of 1/(1+x^2) has a wide range of applications:

• Approximating the function: This series can be used to approximate the value of 1/(1+x^2) for small values of x. The more terms we include, the more accurate our approximation becomes.
• Integrating the function: The series can be integrated term-by-term to obtain the Taylor series of the arctangent function:

  arctan(x) = x - x^3/3 + x^5/5 - x^7/7 + ...
  

• Understanding the behavior: The series reveals the function's behavior near x=0, showing that it is an even function (symmetric about the y-axis) and that it oscillates as x increases.

Additional Notes (Inspired by Github discussions)

• Convergence: The Taylor series of 1/(1+x^2) converges for all values of x within the interval -1 < x < 1. This means that the series can accurately represent the function within this range.
• Relation to the arctangent function: As mentioned, integrating the Taylor series term-by-term gives us the Taylor series of the arctangent function. This highlights a crucial relationship between these two functions and demonstrates the power of Taylor series in connecting different mathematical concepts.

Credit: This article draws inspiration from discussions on the Taylor series of 1/(1+x^2) found on Github. Special thanks to contributors who shared their insights and helped to clarify the concepts.