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Show that the function $ f(x) = \sum_{n = 0}^{\infty} \frac {(-1)^n x^{2n}}{(2n)!} $is a solution of the differential equation $ f''(x) + f(x) = 0 $

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$f(x)+f^{\prime \prime}(x)=0$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 9

Representations of Functions as Power Series

Sequences

Series

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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.

04:35

Show that the function

03:01

Show that the function…

02:33

(a) Show that the function…

02:27

06:59

Show that the function rep…

02:02

01:32

03:18

so show that function is a solution. That the differential equation after what dash first after equals zero. Okay, so we're first going to find what it is the ruler of FX. So this could be in front zero to infinity would differentiate. This part is not You want off in extra power if two and minus one and two months. One tutorial. So this the first you're over there, Okay. And of course, is from zero. It's not from serious. From what? Because the first time within this zero it becomes a constant so and stuff on one. And secondly, relieve a bit is the first duty of of the derivative of X and they just going to be so when in this one. And it's not a constant, though. So Okay, so it's really stewed from one and next You'LL want to help in extra power to minus two and two minutes to Victoria. Yeah, and heel into the treek. If we change so an x one and chemicals and minus one piece because to zero if we change at Michel's tube zero community, so much want the power. So in the coast, plus one, this is a plus one and extra power to an over two and a pictorial. No. So this is just nothing won Tons and from zero to infinity, eh? That was turf. I'm and extra cult Teo and over two in Factorial. So is this just nothing but that? Okay, so we have shown net a double dash. Now, Which of us a devilish first ethical siro. So that's the solution of the equations.

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