Alternating Harmonic Series sums to ln 2

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Theorem

The alternating harmonic 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 for $\ln \left({1 + x}\right)$:

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

and setting $x = 1$ gives:

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

The fact that it does indeed converge is shown in Alternating Harmonic Series is Conditionally Convergent.

$\blacksquare$


Sources