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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^6} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 5

Alternating Series

Sequences

Series

Missouri State University

Oregon State University

University of Michigan - Ann Arbor

University of Nottingham

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.

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Approximate the sum of the…

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05:11

Find the sum of the series…

01:05

approximate this sum up to four decimal places. Now the first thing is, I see that the Siri's is alternating and therefore I should identify the beyond the positive part of the term in this case, just one over into the sixth since since alternating I know by the theorem in section eleven point five. This alternating Siri's estimation there, um that they ear is bounded above when the heroine, using and in terms, is bounded above by being plus one. So this means we want the smallest end. Such that bien is less than five times will hear will do ten to the minus five. So this is the last number that I just pulled out here. This is just to ensure that were correct. That's a four decimal places. Two. So here, let's go in and solve that for end. So we know we have one over into the six less than five times ten to the minus five. So here we could go to a copulate or just do trial in here. But the first end that makes this true is an equal six. So we need at least six. So this tells us that we need toe Add just five terms because again, this is coming from the difference. Here was the BM plus one and the B end. So the ear is SN, but over here we have in plus one. So the n plus one is six hoops and plus one equals six implies and equals five. So going to the next page, the entire song, oops and plus one over into the six is approximately equal to the sum when just using five does Mel's, that's s five. And going into Wolfram, we just go to the four decimal places here and we have nine, eight, five, and then round up one that will just give us six, nine, eight, five, six. So that is our answer that is correct up to four decimal places, and that's our final answer.

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