Category: 12 HL Homework

Use any of the techniques we have discussed in class to determine the convergence/divergence of the following series.

1. \(\displaystyle{\sum_{n=1}^\infty \frac{1}{\sqrt{n}}}\)
2. \(\displaystyle{\sum_{n=1}^\infty \frac{1}{5n}}\)
3. \(\displaystyle{\sum_{n=1}^\infty \frac{n}{2n-\sqrt{n}}}\)

Having now looked at the definition of the limit of a sequence and some associated terms, see if you can show that the sequence defined by

\[u_n=\sqrt{n+1}-\sqrt{n}\]

converges.

Here’s a short question that will give you an opportunity to practice using Euler’s Method. Complete this before our lesson tomorrow.

Use Euler’s Method to find the approximate value of \(f(3)\) if \(y=f(x)\) is the solution to
\[\frac{\textrm{d}y}{\textrm{d}x} = y^2-x\]passing through \((2,1)\). Use a step length of \(0.5\).

(Notice that this is a first-order nonlinear DE that is neither separable hor homogeneous, so we won’t be able to use any methods available in the course to find an explicit solution for this DE.)

In today’s lesson we determined that \(y=\pm\sqrt{x^2+C}-1\), for \(C\in\mathbb{R}\), is the general solution to the differential equation

\[\frac{\textrm{d}y}{\textrm{d}x}=\frac{x}{y+1}\]

As a quick exercise tonight, verify that functions of this form are indeed solutions to the differential equation.

One of the questions that we had a brief look at today is posted below. Our discussion in class dealt with the case for \(p\leq1\), so all we need now to consider is the case where \(p>1\). A comparison test may prove to be difficult, so can you think of another way to establish this result?

For which values of \(p\) does \(\displaystyle{\int_e^\infty \frac{\ln x}{x^p}\textrm{d}x}\) converge?

Following our discussion of improper integrals, complete the questions below prior to the start of our lesson on Tuesday.

Pages 1354–1355 questions 33, 34, 36, 3, 4, 5, 7, 11, 14

L’Hôpital’s Theorem can be used when evaluating limits that have a certain sort of “indeterminate form” (either \(\frac{0}{0}\) or \(\frac{\pm\infty}{\pm\infty}\)). Subject to some other conditions (What are they? Make sure you check that they’re satisfied!), we can use L’Hôpital’s Theorem to calculate limits like

\[\lim_{x\to 0}\frac{\sin x}{x}\]

Use L’Hôpital’s Theorem to find the value of this limit for tomorrow’s lesson. Also, what do you think about the result shown below?

\[\lim_{x\to 0}\frac{\cos x}{x}=\lim_{x\to 0}\frac{-\sin x}{1}=0\]

Complete the following questions for our lesson tomorrow.

Pages 898–901 questions 1, 2, 7, 11, 19, 20

Complete the following questions for the start of our lesson tomorrow.

Complete the assignment attached below for Wednesday, November 9th.

Probability Assignment