Sean Fitzpatrick |
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University of Lethbridge |
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Given the geometric series formula \(\di \sum_{n=0}^\infty x^n = \frac{1}{1-x}\text{,}\) determine a power series representation for \(\di f(x) = \frac{x^4}{(1-x)^3}\text{.}\)
Give an example of a power series whose interval of convergence is \((0,6]\text{.}\)
Use a power series to evaluate the integral \(\di\int_0^2 e^{x^3}\,dx\)
Recall: given a function \(f\) differentiable at \(c\text{,}\) its linear approximation at \(c\) is given by
Higher-order approximations are given by Taylor polynomials:
If a Taylor polynomial is centred at \(0\text{,}\) we call it a Maclaurin polynomial.
The Taylor series of an infinitely differentiable function \(f(x)\) is given by
When \(c\) is zero, we get the Maclaurin series for \(f\text{.}\)
Recall the Lagrange formula for the remainder \(R_n(x)=f(x)-P_n(x)\) obtained when we approximate a function \(f\) by its degree \(n\) Taylor polynomial:
For many common functions, it's not hard to show that \(\lim\limits_{n\to \infty}R_n(x)=0\) for all \(x\) in the radius of convergence of the Taylor series.
Letting \(n\to\infty\) in the equality \(f(x)=P_n(x)+R_n(x)\text{,}\) we get
Give the Taylor series for:
\(f(x) = e^x\text{,}\) centred at \(0\text{.}\)
\(g(x) = \cos(x)\text{,}\) centred at \(0\text{.}\)
\(h(x)=\ln(x)\text{,}\) centred at \(1\text{.}\)
Give the interval of convergence for each one.
We can add, subtract, multiply and divide power series. Substitution works, too. (Sort of.)
In the case of Taylor series, this is equivalent to doing the same to the corresponding functions.
Determine Taylor series for:
\(\di f(x) = \frac{x^3}{1-x}\)
\(\di g(x) = \frac{\sin(x)}{e^x}\)
\(h(x) = \sec(x)\)
\(s(x) = \sin(x^3)\)
We have \(\di (1+x)^\alpha = \sum_{k=0}^\infty \binom{\alpha}{k}x^k\text{,}\) for \(\abs{x}\lt 1\text{.}\)
But note the binomial coefficients are a bit different if \(\alpha\) is not a positive integer:
Find Taylor series for:
\(f(x) = \sqrt{1+2x}\)
\(\di g(x) = \frac{1}{(1+x^2)^2}\)
Many new functions (not expressible in terms of elementary functions) arise as power series, often as solutions to differential equations.
The Bessel functions are named after Friedrich Bessel, who found them as solutions to Kepler's equations. They also show up in problems involving vibrations.
The Bessel function of order zero is given by
The Bessel function of order one is given by
Evaluate the limit:
\(\di \lim_{x\to 0}\frac{\cos(x)-1-x^2}{x^4}\)
\(\di\lim_{x\to 0}\frac{\sqrt{1+x}-1-\frac12 x}{x^2}\)
We saw earlier that given a power series \(\di \sum_{n=0}^\infty a_x(x-c)^n\text{,}\) we have
Compute the integral \(\int_0^{1} \frac{\sin(x^2)}{x^2}\,dx\text{.}\)
What is the error in your approxmiation if you use the first 5 terms in the sum to estimate the value of the integral?
I'll try to fit in a few examples here based on what you asked for on the forum. If you've tuned in live you can suggest others in the comments.
Note: I have no questions planned on direction fields for the test.
Consider the logistic population model \(\di \frac{dP}{dt}=\frac{3}{500}P(93-p)\text{.}\)
For which values of \(P\) is the population increasing? Decreasing?
Find \(P(t)\) given that \(P(0)=48\text{.}\)
Suppose a water tank is being pumped out at 3 L / min . The water tank starts at 10 L of clean water. Water with toxic substance is flowing into the tank at 2 L / min , with concentration \(20t\) g / L at time \(t\text{.}\) When the tank is half empty, how many grams of toxic substance are in the tank (assuming perfect mixing)?
Determine if the series converges:
\(\di \sum_{n=0}^\infty \frac{n}{e^{n^2}}\)
\(\di \sum_{n=0}^\infty \frac{1}{2^n+3n}\)
\(\di\sum_{n=1}^\infty \frac{n^2+1}{\sqrt{n^3+4n+5}}\)
\(\di \sum_{n=1}^\infty\frac{7^k+k}{k!+5}\)