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Show that the series is convergent. How many term…

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Problem 24 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 {( - \frac {1}{3})^n} {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 start off by showing that the syriza's conversion. So here we would use the alternating Siri's test with B N equals one over end, three to the end. This is clearly positive. The limited bien zero. You could see that this term goes to infinity, so the fraction is a whole ghost zero and being is decreasing. This is clearly true because if you just replace and within plus one, that's true. So this thing converges by the alternating Siri's test, so that takes care of the first part of the first sentence. Now we'd like to know how many terms. So what's its end value off the Siri's do we need so that if we take the partial song, then the air when approximating the actual song is less than this fraction over here. So, using the alternating Siri's estimation Tehran, the absolute value of their is less than being plus one. We're here. This is the air when using in terms. So now we want this to be Weston. That's the This is the given number that was given to us. So we'LL find the end that makes this true. So let's what's right This now that's equivalent to end three the end being bigger than two thousand. And then here you would just do some trial in here until you find the first one. If you look at a five not quite large enough. However, when you could say six, this is what we want. This is bigger than the two thousand. So we want and equal six here or really in this case we want and plus one equals six. So that just means and equals five. So we'LL just add five terms is all that is required, and that's our final answer.

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