General Mathematics

    

A Power Series Divergent at Half Pi Yields a Sequence Briskly Convergent to Pi

Authors: Steven Kenneth Kauffmann

The tangent function's Taylor expansion about zero is a series of odd powers with positive coefficients; it converges when the absolute value of its argument is less than half pi, and diverges to positive infinity when its argument equals half pi, so with the aid of the ratio test it is seen that twice the square root of the ratio of its successive coefficients is a sequence which converges to pi. The odd derivatives of the tangent function are polynomials in powers of its square with positive integer coefficients, so a recursion of positive integers can be found from which the coefficients of the series described above may be successively obtained. Its related sequence which converges to pi does so monotonically from above, and appears to refine its approximation to pi by about one significant decimal figure per successive term.

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[v1] 2024-04-02 11:51:48

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