Taylor Series of Combinations of Functions Examples 1
On the Taylor Series of Combinations of Functions page, we use the following Maclaurin series to obtain Taylor series of various functions:
- The Geometric Series: $\frac{1}{1 - x} = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + ...$, for $-1 < x < 1$
- The Derivative of the Geometric Series: $\frac{1}{(1 - x)^2} = \sum_{n=0}^{\infty} nx^{n-1} = x + 2x + 3x^2 + ...$, for $-1 < x < 1$.
- The Antiderivative of the Geometric Series: $-\ln (1 - x) = \sum_{n=0}^{\infty} \frac{x^{n+1}}{n+1} = x + \frac{x^2}{2} + \frac{x^3}{3} + ...$, for $-1 ≤ x < 1$.
- Inverse Tangent Function: $\tan ^{-1} x = \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} x^{2n+1} = x - \frac{x^3}{3} + \frac{x^5}{5} - ...$, for $-1 ≤ x ≤ 1$.
- Euler Exponential Function: $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + \frac{x}{1!} + \frac{x^2}{2!} + ...$, for $(-\infty, \infty)$.
- Natural Logarithm: $\ln (1 + x) = \sum_{n=0}^{\infty} (-1)^{n+1} \frac{x^n}{n}$.
- Sine Function: $\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!}x^{2n+1} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - ...$, for $(-\infty, \infty)$.
- Cosine Function: $\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n} = 1 - \frac{x^2}{2} + \frac{x^4}{4} - ...$, for $(-\infty, \infty)$.
We will now look at some more examples of obtaining Taylor series from the Maclaurin series above.
Example 1
Find a Taylor series representation of the function $f(x) = \cos^2 x$ about $\frac{\pi}{8}$.
Let $t = x - \frac{\pi}{8}$. Then $x = t + \frac{\pi}{8}$ and so:
(1)We use the identity $\cos^2 x = \frac{1 + \cos 2x}{2}$ to get that:
(2)We will now use the sum identity $\cos (x + y) = \cos x \cos y - \sin x \sin y$ to get that:
(3)The Maclaurin series for $\cos t$ and $\sin t$ are respectively:
(4)Therefore, the Maclaurin series for $\cos 2t$ and $\sin 2t$ are respectively:
(6)Substituting these sums into what we obtained earlier and we have that:
(8)