61 lines
844 B
Markdown
61 lines
844 B
Markdown
#Math #Calculus
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Represent function using power series:
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$$
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f(x) = \sum _{n=0}^{\infty} c_n (x-a)^n
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$$
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Find $c_0$
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$$
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c_0=f(a)
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$$
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Take derivative of function
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$$
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\frac d {dx} f(x) = \sum _{n=0}^\infty c_{n+1} (n+1)(x-a)^n
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$$
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Find $c_1$
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$$
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c_1=\frac {d} {dx} f(a)
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$$
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Take second derivative of function
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$$
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\frac {d^2} {d^2x} f(x) = \sum _{n=0}^\infty c_{n+2} (n+1)(n+2)(x-a)^n
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$$
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Find $c_2$
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$$
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c_2=\frac {\frac {d^2} {d^2x} f(a)} {2}
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$$
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Take third derivative of function
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$$
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\frac {d^3} {d^3x} f(x) = \sum _{n=0}^\infty c_{n+3} (n+1)(n+2)(n+3)(x-a)^n
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$$
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Find $c_3$
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$$
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c_3=\frac {\frac {d^3} {d^3x} f(a)} {6}
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$$
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Create general formula for $n$th element of $c$
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$$
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c_n = \frac {\frac {d^n} {d^nx}f(a)} {n!}
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$$
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Create general formula for function as polynomial
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$$
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f(x)=\sum _{n=0}^\infty \frac {\frac {d^n} {d^nx}f(a)} {n!} (x-a)^n
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$$ |