Taylor series for e^x, Single Variable Calculus

We find the Taylor series (or Maclaurin series) for f(x)=e^x by computing the coefficients and the radius of convergence. Since the derivative of 𝑒^𝑥 is 𝑒^𝑥 itself, 𝑓^{(𝑛){(0)=𝑒^0=1. Therefore, each Taylor coefficients 𝑐𝑛 is 1/𝑛!, and the Maclaurin Series for 𝑒^𝑥 is e^x = Σ x^n/n!, beginning with n=0.

Using the "coefficients only" version of the ratio test, we establish that the radius of convergence is infinite, so the series is valid for all real numbers.

We then use this to tackle an integral that we cannot do by hand (in terms of elementary functions). This lecture is part of my course on Single Variable Calculus.

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