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Approximate the sum of the series correct to four decimal places.

$ \displaystyle \sum_{n = 1}^{\infty} \frac {( - 1)^{n-1}}{n 4^n} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 5

Alternating Series

Sequences

Series

Campbell University

Harvey Mudd College

Idaho State University

Boston College

Lectures

01:59

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

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.

01:09

Approximate the sum of the…

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05:11

Find the sum of the series…

it's approximate this some of this alternating serious. It's a four decimal places. So here, if we want to be correct up to four decimal places, we need the air to be less than I'LL put the less than way out here zero point zero zero zero zero five and explain that. So if the ear is lost in this decimal over here, so notice that if you're less than this, then when you round off the this number in the fifth spot is not large enough, Teo increase the four decimal. That means that we're currently four decimal places. That's the reason for this number, however, sense for dealing with the alternating Siri's the alternating Siri's estimation. Dirham says that the air, when approximating the sum by using in terms and the partial song, is less than being plus one or being is just the absolute value of a M. So bn is just one over end times for the and power. So we want this to be less than zero point zero zero zero zero five. That turns out to be true. Using a calculator, you could see that you want and to be six or more so here we have n plus one bigger than or equal to six so and bigger than equal So five. That means that in our approximation that we should just be using five terms. So here they're approximation will just be sn or as five, actually, because here, that's the number five is approximately so this you will go to the calculator. I'LL write this out in the exact form. So this is just the sum from one to five negative one and minus one over and four to the end and then going toe a calculator and plugging this in way into two and then three two. So that's our approximation of the infinite sums of four decimals, and that's our final answer.

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