Today we introduced Taylor series and Maclaurin series. A Taylor series is a power series for a function \(f(x)\) that can be calculated by finding all of the derivatives of the function when \(x=c\):
\[f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(c)}{n!} (x-c)^n\]
In this formula, \(f^{(n)}(c)\) is the n-th derivative of \(f\) at the point \(x=c\) (recall that \(c\) is the center of the power series. Usually \(c=0\)). When the center is \(c=0\), we call the series a Maclaurin series sometimes (but it is still a Taylor series too).
We did the following examples in class:
Find the Maclaurin series for \(f(x) = \sin x\) by making a table of derivatives and finding the pattern.
Find the Maclaurin series for \(f(x) = e^x\) by making a table of derivatives and finding the pattern.
Find the Maclaurin series for \(f(x) = \cos x\) by differentiating the Maclaurin series for \(\sin x\).
Find the radius of convergence for the Maclaurin series for \(e^x\) using the formula \(R = \displaystyle \lim_{n \rightarrow \infty} \frac{|a_n|}{|a_{n+1}|}\).
We finished by talking about how the n-th partial sum \(S_n\) of a series doesn’t exactly equal the infinite sum \(S_\infty\). We call the difference the error in the partial sum. For some types of series, we have formulas that let us know how bad the error might get. The simplest error formula is the one for alternating series:
For an alternating series \(\displaystyle \sum_{n=0}^\infty (-1)^n b_n\), the error in a partial sum is always smaller than the next term that wasn’t included:
\[|\text{Error}| = |S_\infty - S_n| \le b_{n+1}.\]
We used this formula to solve the following problems:
\(\sin(1) \approx 1 - \frac{1}{3!} + \frac{1}{5!} - \frac{1}{7!} + \frac{1}{9!} - \frac{1}{11!}.\) Estimate the worst case error in this approximation.
How many terms of the alternating harmonic series \(1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \ldots\) would you need to add to get a partial sum with an error less than \(\frac{1}{100}\)?
Today we went over the Midterm 3 Review Problems and Homework 10. I also handed out a summary of what we’ve covered on infinite series:
We also talked about how power/Taylor series let you calculate impossible integrals. We did this example in class:
Find a Maclaurin series for \(\dfrac{\sin x}{x}\).
Find the integral \(\dfrac{\sin x}{x} \, dx\) by using the Macluarin series.
Find the area under the curve \(y = \dfrac{\sin x}{x}\) from \(x=0\) to \(x=1\). Express your answer as an infinite series.
The answer we got was the infinite series \(\displaystyle 1 - \frac{1}{3 \cdot 3!} + \frac{1}{5 \cdot 5!} - \frac{1}{7 \cdot 7!} + \ldots\)