# Improving the convergence of a slowly convergent series

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.

© SolutionLibrary Inc. solutionlibary.com November 29, 2020, 1:38 am 9836dcf9d7 https://solutionlibrary.com/math/numerical-analysis/improving-the-convergence-of-a-slowly-convergent-series-jguk