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Use the Taylor series for $\ln x$ at $x=1$ to find the sum of the alternating harmonic series:

$$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots$$

$\cfrac{1}{2}\ln 2$

$\cfrac{1}{2}\ln 3$

$\ln 2$

$1$

$\ln 3$