Understanding Pi through Taylor Series: A Detailed Exploration

The value of pi is a constant mathematical value, approximately 3.1415926535, irrespective of the formula used to compute it. pi can be approximated using various methods, including the Taylor series, which provides a powerful tool for understanding its nature and approximation.

Which Taylor Series?

Though there isn't a single Taylor series for pi, there are several different series that can be used to approximate it. The choice of series might vary based on the function selected. For example, the Maclaurin series for the inverse tangent function, arctan(x), can be applied to approximate pi.

Maclaurin Series for arctan(x)

The Maclaurin series for arctan(x) is given by:

$$arctan(x) sum_{n0}^{infty} frac{(-1)^n x^{2n 1}}{2n 1} x - frac{x^3}{3} frac{x^5}{5} - frac{x^7}{7} ldots$$

This is also known as the Leibniz formula for pi when x 1.

$$arctan(1) frac{pi}{4} sum_{n0}^{infty} frac{(-1)^n}{2n 1} 1 - frac{1}{3} frac{1}{5} - frac{1}{7} ldots$$

Thus, the value of pi can be approximated using this series:

$$pi sum_{n0}^{infty} frac{4(-1)^n}{2n 1} 4 - frac{4}{3} frac{4}{5} - frac{4}{7} ldots$$

While this series is simple due to its regular pattern, it converges quite slowly, as demonstrated by the term counts needed to achieve specific decimal places: 25 for the first digit, 627 for the second, 2454 for the third, and 136,120 for the fourth.

Newton's Series

isaac Newton developed a more efficient series for computing pi. He used a specific Maclaurin series for the function:

$$frac{xsqrt{1-x^2}arcsin(x)}{2}$$

The series he found is:

$$frac{xsqrt{1-x^2}arcsin(x)}{2} -sum_{i0}^{infty} frac{(2i!)x^{2i-1}}{(i!)^24^i(4i^2-1)} x - frac{x^3}{6} - frac{x^5}{40} - frac{x^7}{112} - ldots$$

When x 1, the series simplifies to:

$$frac{pi}{4} -sum_{i0}^{infty} frac{(2i!)}{(i!)^24^i(4i^2-1)} 1 - frac{1}{6} - frac{1}{40} - frac{1}{112} - ldots$$

Thus, the value of pi can be approximated as:

$$pi -sum_{i0}^{infty} frac{(2i!)}{(i!)^24^{i-1}(4i^2-1)} 4 - frac{2}{3} - frac{1}{10} - frac{1}{28} - ldots$$

This series is much faster to converge: it takes only three terms to find the first decimal digit, 13 terms for the second, and 95 terms for the third.

An Even Faster Series

Newton also discovered a second series when x 1/2.

$$frac{sqrt{3}}{8}frac{pi}{12} -sum_{i0}^{infty} frac{(2i!)}{2(i!)^216^i(4i^2-1)} frac{1}{2} - frac{1}{48} - frac{1}{1280} - frac{1}{14336} - ldots$$

Thus, the value of pi can be approximated as:

$$pi -frac{3sqrt{3}}{2} - sum_{i0}^{infty} frac{6(2i!)}{(i!)^216^i(4i^2-1)} -frac{3sqrt{3}}{2} - frac{6}{1} - frac{1}{4} - frac{3}{320} - frac{3}{3584} - ldots$$

This series converges incredibly fast, as it takes only five terms to find the fifth decimal digit and 14 terms for the tenth digit. However, it involves the calculation of sqrt(3), which requires a separate approximation.

Conclusion

The Taylor series provide a theoretical understanding and practical approach to approximate the value of pi. From the slow convergence of pi as given by the Leibniz series to the rapid convergence of Newton's series, each method showcases the elegance and complexity of the mathematical constants that govern our world.

References

Veritasium’s video: [Providing a link to the Veritasium video about Newton's series, if available online]