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Find the Maclaurin series for the functions.$$\sinh x=\frac{e^{x}-e^{-x}}{2}$$

$$\sum_{n=0}^{\infty} \frac{x^{2 n+1}}{(2 n+1) !}$$

Calculus 2 / BC

Chapter 10

Infinite Sequences and Series

Section 8

Taylor and Maclaurin Series

Series

Missouri State University

Campbell University

Harvey Mudd College

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.

14:11

In mathematics, the partial sums of a series are the sums of all terms of the series except possibly the first and last.

02:07

Find the Maclaurin series …

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For the maclaren series of the function of x, which is equal to e 2 x minus e to negative x over 2 point, so first we used maclaren's series over the function e 2 x. This is equal to some and from 0 to infinity x, to the power n over n factorial and replace x by 2 x. We have the michelin series of the function to negative x is some from 0 to infinity negative 1 to the power n times x. To n over n factorial, then cinch of x is equal to series and, from 0 to infinity x, over n factorial minus negative 1 to n x, o n over n bacteria, which is equal to it. Here we need to multiply the half 1 half times sum and from 0 to infinity here, 1 minus negative 12 reunite to 0. This is 0. The n is equal to 1, is equal to 2 times x over 1 and then, as 0 plus equal to 3 x, 2 x, cube over x 2 over 3 factorial, plus to the door, which is equal to x, plus x, cubed over 3 factorial plus 5. Over 5 factorial plus dotto, which is equal to some and from 0 to infinity x, to the power 2 n plus 1 over 2 n plus 1 factorial.

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