# Power Series Expansion for Logarithm of 1 + x

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## Theorem

The Newton-Mercator series defines the natural logarithm function as a power series expansion:

 $\displaystyle \map \ln {1 + x}$ $=$ $\displaystyle \sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} \frac {x^n} n$ $\displaystyle$ $=$ $\displaystyle x - \frac {x^2} 2 + \frac {x^3} 3 - \frac {x^4} 4 + \cdots$

valid for all $x \in \R$ such that $-1 < x \le 1$.

### Corollary

 $\displaystyle \ln \paren {1 - x}$ $=$ $\displaystyle -\sum_{n \mathop = 1}^\infty \frac {x^n} n$ $\displaystyle$ $=$ $\displaystyle -x - \frac {x^2} 2 - \frac {x^3} 3 - \frac {x^4} 4 - \cdots$

valid for $-1 < x < 1$.

## Proof

From Sum of Infinite Geometric Progression, putting $-x$ for $x$:

$(1): \quad \displaystyle \sum_{n \mathop = 0}^\infty \paren {-x}^n = \frac 1 {1 + x}$

for $-1 < x < 1$.

From Power Series Converges Uniformly within Radius of Convergence, $(1)$ is uniformly convergent on every closed interval within the interval $\openint {-1} 1$.

From Power Series is Termwise Integrable within Radius of Convergence, $(1)$ can be integrated term by term:

 $\displaystyle \int_0^x \frac 1 {1 + t} \rd t$ $=$ $\displaystyle \sum_{n \mathop = 0}^\infty \int_0^x \paren {-t}^n \rd t$ $\displaystyle \leadsto \ \$ $\displaystyle \map \ln {1 + x}$ $=$ $\displaystyle \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{n + 1} } {n + 1}$ Primitive of Reciprocal and Integral of Power $\displaystyle \leadsto \ \$ $\displaystyle \map \ln {1 + x}$ $=$ $\displaystyle \sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} \frac {x^n} n$ letting $n \to n - 1$

$\blacksquare$