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    Improving the convergence of a slowly convergent series

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    We want to evaluate log(2) by inserting x = 1 in the series:

    log(1+x) = x - x^2/2 + x^3/3 - x^4/4 + x^5/5 - x^6/6 + x^7/7 - x^8/8 + x^9/9 - x^10/10 +...

    and take the first ten terms. But the result is rather poor. Show that by taking a linear combination of this truncated series with that of the function 1/(1+x) that eliminates the coefficient of x^10, one obtains a much better result.

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    Solution Preview

    The series expansion:

    log(1+x) = x - x^2/2 + x^3/3 - x^4/4 + x^5/5 - x^6/6 + x^7/7 - x^8/8 + x^9/9 - x^10/10 +... (1)

    converges very slowly to log(2) for x = 1. The first ten terms sum to 0.6456 which is quite a poor estimate for log(2), with an error of approximately 0.0475. The slow convergence is caused by the presence of a branch point singularity at x = -1. One can improve the convergence by taking a linear combination of the series with that of another function that also has a ...

    Solution Summary

    The Solution explains in detail how this method of adding another series to improve the convergence works.