The Integral Test

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.

The Limit Comparison Test

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

Calculus Option Test 1

We’ll have our first test on the Calculus Option topics on Thursday, March 9th.

This test will cover material that includes differential equations, important theorems and definitions in Calculus (Rolle’s Theorem, the Mean Value Theorem, continuity and differentiability, etc.), and l’Hôpital’s Rule.

The following questions will be useful.

page 1471–1474 questions 1, 4, 10, 16, 19, 22–27

page 1436 questions 9, 17

page 1354 questions 4, 6, 14, 15