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Theorem: Maclaurin Series of ln(1 + x)

The Maclaurin series of ln⁡(1+x)\ln(1+x) is ∑n=1∞(−1)n−1nxn\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}x^n, and

ln⁡(1+x)=∑n=1∞(−1)n−1nxn=x−x22+x33−x44+⋯for −1<x≤1,\ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}x^n = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \qquad \text{for } -1 < x \le 1,

while the series diverges for every other real xx.

22 viewsMathematics · Single Variable Calculus
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