Mathematical Methods in the Physical Sciences/Chapter 1/Section 5

Overview

Introducing the preliminary test!

Notes

• The preliminary test of the convergence of a series says if <math>\lim_{n \to \infty}a_n \ne 0</math>, then the series diverges.
• The preliminary test say nothing about convergence.
• For some reason, you'll get a lot of mileage out of L'Hopital's Rule.

Problems

#1

<math> \begin{align} \lim_{n \to \infty}{(-1)^n n^2 \over n^2-1} &= \lim_{n \to \infty}(-1)^n \lim_{n \to \infty}{n^2 \over n^2-1} \\ &= \lim_{n \to \infty}(-1)^n \lim_{n \to \infty}{2n \over 2n} \text{ (L'Hopital's Rule)} \\ &= \lim_{n \to \infty}(-1)^n \\ \end{align} </math>

Series diverges.

Cautionary Note: If you consider pairs of terms, you can probably see that pairs tend to 0. However, we're not taking the sum of pairs of terms, we're taking the sum of the terms one by one. Near the "end" of the series, the sum will alternate between some value and some value -1. Thus, it has no sum, and is not convergent.

#2

<math> \begin{align} \lim_{n \to \infty} {\sqrt{n+1} \over n} &= \lim_{n \to \infty} {{1 \over 2\sqrt{n+1}} \over 1} \text{ (Hello, L'Hopital!)} \\ &= \lim_{n \to \infty} {1 \over 2(n+1)^{1 \over 2}} \\ &= \lim_{n \to \infty} {0 \over (n+1)^{3 \over 2}} \text{ (More L'Hopital)} \\ &= 0 \end{align} </math>

We don't know if it is convergent or not.

Cautionary Note: The preliminary test says nothing about convergence. It only states that series with non-zero limits are divergent, that is all.

#3

<math> \begin{align} \lim_{n \to \infty} {n + 3 \over n^2 + 10n} &= \lim_{n \to \infty} {1 \over 2n + 10} \text{ (L'Hopital)} \\ &= \lim_{n \to \infty} {0 \over 2} \text{ (Guess who?)} \\ &= 0 \end{align} </math>

We don't know if it is convergent or not.