Infinite Series 9 Copyright © Cengage Learning. All rights reserved.

The Integral Test and p-Series Copyright © Cengage Learning. All rights reserved. 9.3

Use the Integral Test to determine whether an infinite series converges or diverges. Use properties of p-series and harmonic series. Objectives

Srīnivāsa Rāmānujan (22 December 1887 – 26 April 1920) was an Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series and continued fractions. Ramanujan's talent was said by the English mathematician G.H. Hardy to be in the same league as legendary mathematicians such as Gauss, Euler, Cauchy, Newton and Archimedes and he is widely regarded as one of the towering geniuses in mathematics.

Born to a poor Brahmin family, Ramanujan first encountered formal mathematics at age 10. He demonstrated a natural ability, and was given books on advanced trigonometry written by S. L. Loney. He mastered them by age 12, and even discovered theorems of his own, including independently re-discovering Euler's identity. He demonstrated unusual mathematical skills at school, winning accolades and awards. By 17, Ramanujan conducted his own mathematical research on Bernoulli numbers and the Euler– Mascheroni constant. He received a scholarship to study at Government College in Kumbakonam, but lost it when he failed his non-mathematical coursework. Ramanujan joined another college to pursue independent mathematical research, working as a clerk in the Accountant- General's office at the Madras Port Trust Office to support himself. In 1912–1913, he sent samples of his theorems to three academics at the University of Cambridge.

Only Hardy recognized the brilliance of his work, subsequently inviting Ramanujan to visit and work with him at Cambridge. He became a Fellow of the Royal Society and a Fellow of Trinity College. Ramanujan was a devout Hindu and would not eat meat, nor would he eat food prepared in any containers that had ever held any meat. Also, his wife who had stayed behind in India, told of times when she had to literally put food in his mouth as he would forget to eat for days, entranced by his “sums.” Without his wife’s care and with limited suitable food available in England, Ramanujan died of malnutrition, and possibly liver infection, in 1920 at the age of 32.

For the next few lessons, we will be concerned with testing many different types of series for convergence. We will not be particularly concerned with finding the sums of the series that do happen to converge. If that bothers you, you may be an engineer. Feel free to compute the infinite sums of convergent series in your copious spare time. …just don’t forget to eat…

The integral Test is one of many tests for convergence. It is elegant and simple, but unfortunately has limited application. Example: Integral Test is appropriate. Example: Integral Test is NOT appropriate.

The terms of the series can form an upper sum. If the integral diverges…so does the “larger” series. The same terms can also form a lower sum.

The terms of the series can form an upper sum. If the integral diverges…so does the “larger” series. The same terms can also form a lower sum. If the integral converges…so does the series.

Because the series can be drawn as either an upper or lower sum, we can conclude that whatever the integral does (converges or diverges), the series does too. To use the integral test, all terms of the series must be positive and decreasing over the long run. You also need to know how to integrate the expression

Integral Test either both converge or both diverge.

Ex: #1 Apply the integral test: Since the integral diverges, so does the series. Terms positive? Decreasing? Can we integrate it?

Ex: #2 Apply the integral test: Since the improper integral converges, so does the series. Terms positive? Decreasing? Can we integrate it?

Ex: #3 Decide if the series converges: Since the integral converges, so does the series. Terms are positive, but are they decreasing?

Ex: #4 Apply the integral test to the decreasing positive series: Since the integral diverges, so does the series. The only thing that can make these problems difficult is if the integral itself is difficult.

Ex: #5 You Try: Apply the Integral Test to the series Solution: The function is positive and continuous for To determine whether f is decreasing, find the derivative.

Solution So, f'(x) 1 and it follows that f satisfies the conditions for the Integral Test. You can integrate to obtain So, the series diverges.

p-Series and Harmonic Series

A second type of series has a simple arithmetic test for convergence or divergence. A series of the form is a p-series, where p is a positive constant. For p = 1, the series is the harmonic series. p-Series and Harmonic Series

A general harmonic series is of the form In music, strings of the same material, diameter, and tension, whose lengths form a harmonic series, produce harmonic tones. p-Series and Harmonic Series

p-Series Any series of the form where p is a real number.

Ex: #6– Convergent and Divergent p-Series Discuss the convergence or divergence of (a) the harmonic series and (b) the p-series with p = 2. Solution: a.From Theorem 9.11, it follows that the harmonic series diverges. b. From Theorem 9.11, it follows that the p-series converges.

Ex: #7 Determine whether the Series converges or diverges: The series diverges. This series is similar to the harmonic series. If its terms were larger than those of the harmonic series, we would expect it to diverge. Because its terms are smaller, we are not sure what to expect. The function is positive and continuous for x>2. To determine whether f is decreasing, find its derivative.

Ex: #16 Use the integral test:

If we stop evaluating a convergent positive series after adding n terms, the difference between the true infinite sum and our partial sum is called the error or remainder. In the case where all the terms are positive, the error (or remainder) will be positive. Here’s a representation of the series (Riemann Sum) and it’s related integral for n large.

Ex: The minimum number of terms needed is 32.

AP Practice

Day 1: Pg odd Day 2: Pg odd, odd Day 3: MMM pg. 196 (9.2), 199 (9.3)