abMathematics

Have a look at the following questions from option topic (Series and Differential Equations) past papers shown below.

November 2011, questions 2 and 4.

November 2012, question 3.

We will discuss these in our next lesson.

One nice feature of the integral test is that—when it can be applied to show that a given function converges—we also get an easy way to find a bound on the size of the error term.

In other words, we can use a partial sum to find an approximation of the actual value of a series while at the same time knowing how far off we are (at worst) from the true value of the sum of that series.

Complete the question below for tomorrow’s lesson.

Consider the \(p\)-series with \(p=3\).

• Find a general expression for a bound on the error term when \(S_k\), for some \(k \in \mathbb{Z}^+\), is used to approximate the sum.
• Find the number of terms required to approximate the value of this series to within 0.0005.
• Find the value of \(\displaystyle{\sum_{n=1}^\infty \frac{1}{n^3}}\), accurate to three decimal places.

Now that we’ve looked at the unit circle definitions of the trigonometric functions, complete the following questions.

Complete the following questions for our next lesson.

Pages 286–288, questions 1, 2, 3, 8, 10, 12, 25, 28, 29, 34, 35, 38, 40

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.)