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Every function can be written as an infinite-order polynomial
cn=f(n)(0)n!c_n = \frac{f^{(n)}(0)}{n!}cn=n!f(n)(0)
f(x)=∑n=0∞f(n)(0)n!xnf(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^nf(x)=∑n=0∞n!f(n)(0)xn
f(0)f(0)f(0)
f′(0)f'(0)f′(0)
1+x+x221 + x + \frac{x^2}{2}1+x+2x2
x−x22+x33+⋯x - \frac{x^2}{2} + \frac{x^3}{3} + \cdotsx−2x2+3x3+⋯
Factor out the constant: ln[2(1+x2)]=ln2+ln(1+x2)\ln[2(1 + \frac{x}{2})] = \ln 2 + \ln(1 + \frac{x}{2})ln[2(1+2x)]=ln2+ln(1+2x)
Write as a product of known expansions
Ignore all powers greater than 2
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