Newton-Mercator Series/Examples/2

Theorem
The alternating horizontal series converges to the natural logarithm of $2$:
 * $\displaystyle \sum_{n \mathop = 1}^\infty \left({-1}\right)^{n - 1} \frac 1 n = \ln 2$

Proof
From Power Series Expansion of Logarithm Function:
 * $\displaystyle \ln \left({1 + x}\right) = \sum_{n \mathop = 1}^\infty \left({-1}\right)^{n - 1} \frac {x^n} n$

and setting $x = 1$ gives:


 * $\displaystyle \sum_{n \mathop = 1}^\infty \left({-1}\right)^{n - 1} \frac 1 n = \ln 2$

Note that $1$ is on the radius of convergence of $\displaystyle \sum_{n \mathop = 1}^\infty \left({-1}\right)^{n - 1} \frac {x^n} n$.

The fact that it does indeed converge is shown in Sum of Reciprocals is Divergent.