“Key Idea 32” (From the text: Apex Calculus) gives the \(nth\) term of the Taylor series of common functions. In Exercises 3 - 6, verify the formula given in the Key Idea by finding the first few terms of the Taylor series of the given functionon and identifying a pattern.
\[f(x) = e^x; \space c=0\]
The first 4 terms (and generalization), as per Key Idea 32 is as follows:
\[ 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^n}{n!} \]
Now we look at the first few terms and we see that they’re all equal to 1 \[ e^0 = 1 \\ \frac{d}{dx} \big(e^x)(0) = 1\\ \frac{d^2}{dx^2} \big(e^x)(0) = 1\\ \frac{d^3}{dx^3} \big(e^x)(0) = 1\\ \frac{d^4}{dx^4} \big(e^x)(0) = 1\\ \frac{d^n}{dx^n} \big(e^x)(0) = 1\\ \]
So the Taylor series looks like this:
\[ 1+x+\frac{1}{2}x^2+\frac{1}{3}x^3+\frac{1}{4}x^4+\frac{1}{n}x^n \]
And as such, the series (or pattern) is:
\[ \sum_{n=0}^{\infty} \frac{x^n}{n!} \]