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Use the power series for $ tan^{-1} x $ to prove the following expression for $ \pi $ as the sum of an infinite series: $ \pi = 2 \sqrt 3 \sum_{n = 0}^{\infty} \frac {(-1)^n}{(2n + 1) 3^n} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 9

Representations of Functions as Power Series

Sequences

Series

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

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.

04:19

Use the power series for t…

05:12

Find the sum of each serie…

02:01

$$\begin{aligned}&…

06:07

help on question

01:07

Find the sum of the infini…

The problem is: use the power series for tending the inverse x to prove the following expression for pi, as the form of an infinite series. So first go, we have up. Tangent x is equal to some from 0 to infinity negative 1 to the power of n times x, 22 n plus 1 over 2 n plus 1 point absolutist last of 1. Now the light x is equal to square root of 3 over 3 pi, or this is 1 over root of 3 and we have a tangent 1 over root to 3. So this is equal to pi over 6 and then we use this formula we have. This is also equal to and from 0 to infinity, negative 1 to the power of n times 1 over 2 n plus 1 times to 3 to the power 2 n plus 1, which is equal to 1 over 3 times sum from 0 to infinity negative 1 To the power of n times 1 over 2 n plus 1 times 3 to the power of n, then we have pi is equal to 6 times 1 over to it 3 times the sum from 0 to infinity negative 12 times 1 over 2 n plus 1 Times 3 to m point notice that 6 times 1 over root of 3. This is equal to 2 times 0 to 3 point, so pi is equal to 2 times root of 3 times this se.

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