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A Maclaurin series is an infinite series that is a method to calculate an approximation of a function, f(x), for x values close to zero, given that the successive derivatives of f(x) at zero can be calculated.
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\n" ); document.write( "For this problem, f(x) = e^(kx), where k is a real number
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\n" ); document.write( "1) Calculate the derivatives for f(x)
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\n" ); document.write( "f'(x) = ke^(kx), f''(x) = k^2e^(kx), ,,,,, f^n(x) = k^n * e^(kx)
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\n" ); document.write( "2) At x = 0,
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\n" ); document.write( "f(0) = 1, f'(0) = ke^0 = k, f''(0) = k^2, .... f^n(0) = k^n
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\n" ); document.write( "Using the definition of the Maclaurin for e^(kx) is
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\n" ); document.write( "e^(kx) = summation from n=0 to +infinity of f^n (0) (x^n/n!) =
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\n" ); document.write( "1 +kx +(k^2 * x^2 / 2!) +(k^3 * x^3 / 3!) +.... = summation n=0 to +infinity of (k^n * x^n / n!)
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