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Show that if we want to approximate the sum of the series $ \sum_{n = 1}^{\infty} n^{-1.001} $ so that the error is less than 5 in the ninth decimal place, then we need to add more than $ 10^{11,301} $ terms!

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$1.07 \times 10^{11,301}$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 3

The Integral Test and Estimates of Sums

Sequences

Series

Missouri State University

Oregon State University

Harvey Mudd College

Baylor University

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.

05:32

Show that if we want to ap…

00:48

03:05

Use the sum of the first 1…

02:49

03:03

03:30

this problem will show that if we want to find a partial some to approximate this infinite sum here such that the ear is less than this decimal here that will be eight zeros and then a five in the ninth spot. Then we'LL show that we'LL need to take and larger than this quantity over here. So this is our am So that means we should find off of X by just replacing and with X And then here X is bigger than or equal to one. And the reason for pointing that out is if you use in terms then from the section we know that the air is bounded above and we want thiss which is equal to one. You can go ahead and replace F with the formula up here and we would like this here to be less than the given quantity. So instead of writing this each time, let me just go ahead and denote this by some water, eh? And now this will go ahead and saw friend. So that means that we should just go ahead and use the power rule here and integrate this and that inequality is equivalent to this inequality here and we could just keep rewriting this and then this is equivalent to. So now, finally, to solve this for end will raise both side to the one thousand power that'LL cancel out this here and that'LL leave us with n larger than one thousand over, eh? All to the one thousand color and going to the calculator we get and we can see here that will need more than this many terms because this larger this numbered out here is larger and that completes the proof.

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