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.

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