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Use a power series to approximate the definite integral to six decimal places.$ \int^{1/2}_0 \arctan (x/2) dx $

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$\approx 0.061865$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 9

Representations of Functions as Power Series

Sequences

Series

Campbell University

University of Michigan - Ann Arbor

Idaho State 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.

02:57

Use a power series to appr…

03:34

04:27

04:28

Okay. Use the power series to approximated. Definitely grow to six decimal places. All right, we're gonna first expand. This function so equals two arc tangent. Half of X equals two. So head of Ike's minus have access to the power of three over three and plus some hierarchy. Term of X cube. So which is too small to be related in this? In this difference, the X and this is going to be Is this going to be, uh, X square over four, minus extra power four over 96. Last some higher on the term. So we can just even more. This part. Oh, and so it's from one half. So we're talking one half to dysfunction and minus these values at zero. Okay, Okay. Let me first check this, uh, is that if the correct So we take the derivative of it. So it's gonna be, uh, have axes, okay? And the rest of this is going to be four and six of 24 on the Terminator and execute on the numerator yet, as she's definitely correct. Okay. And we're gonna we're gonna evaluate this. This difference. We just plug in one half and it becomes to zero point 254 and 0.5 to power 4/96. So the final answer is zero point 061849 and we meet this requirements six decimal places. 12346 All right.

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