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Show that the series is convergent. How many term…

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Problem 25 Easy Difficulty

Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy?

$ \displaystyle \sum_{n = 1}^{\infty} \frac {( - 1)^{n-1}}{n^2 2^n} (|error| < 0.0005) $

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Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 11

Infinite Sequences and Series

Section 5

Alternating Series

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Lectures

01:59

Series - Intro

In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a finite sequence of real numbers is called a finite series. The sum of an infinite sequence of real numbers may or may not have a well-defined sum, and may or may not be equal to the limit of the sequence, if it exists. The study of the sums of infinite sequences is a major area in mathematics known as analysis.

02:28

Sequences - Intro

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first "n" natural numbers (for a finite sequence). A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers.

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Problem 36

Video Transcript

Let's show that the syriza's conversion since it's alternating Siri's with continues the alternating Siri's test. So this requires that first, we define being to just be the positive par of the term. And there's three conditions that have to be satisfied. The first one is that beings positive. That's true. The next one is that the limit of D n A as and goes to infinity a zero. That's true. If you take the limit of this term up here, the numerator is just one, but the denominator goes to infinity so that fraction will go to zero, and the last condition is that the beyond sequence is decreasing. To see that this is true. Just go ahead and replace and with n Plus One, and this is definitely less than bien. The reason this is true is because the denominator on the left is larger than the denominator on the right. The larger the denominator, the smaller the freshen. So therefore, the Siri's will converge by the alternating Siri's test. So that's the first part of this question now, since we showed it converges by the alternating Siri's test, we should go ahead and use the alternating Siri's estimation there, Um, also in eleven point five this section. This tells us that the ear an absolute value one of using in terms to approximate the infinite sum is less than or equal being plus one. And we want that to be less than zero point zero zero zero five. Of course, the next cage. So we want if we want to solve this for n so equivalently two to the end and squared is bigger than two thousand. And here the first time this is true is when and his larger six or more. However we wanted we wanted to be in plus one to be less than zero point zero zero zero five. So, really, in our case, we should have and plus one bigger than equal to six. So that means and bigger than or equal to five. So we should choose five terms, and that's our final answer.

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