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\(f(x) = e^x\) at \(c = 0 \implies p_n(x) = 1 +x + \frac{1}{2} x^2 + \frac{1}{6}x^3 + \cdots + \frac{1}{n!} x^n\).
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\(f(x) = \ln(1+x)\) at \(c = 0 \implies p_n(x) = x - \frac {x^2} 2 + \frac{1}{3} x^3 - \cdots \frac{(-1)^n}{n} x^n\)
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\(f(x) = \cos x\) at \(c = 0 \implies p_{2n}(x) = 1 - \frac{1}{2} x^2 + \frac{1}{24}x^4 - \cdots + \frac{(-1)^n}{(2n)!} x^{2n}\)
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\(f(x) = \sqrt x\) at \(c = 1 \implies p_3(x) = 1 + \frac 1 2 (x-1) - \frac 1 8 (x-1)^2 + \frac{1}{16} (x-1)^3\)