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(a) Show that the function $ f(x) = \sum_{n = 0}^{\infty} \frac {x^n}{n!} $is a solution of the differential equation $ f'(x) = f(x) $(b) Show that $ f(x) = e^x. $

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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

Oregon State University

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.

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(a) Show that the function…

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So first we're gonna show the function. F packs is a solution to the differential equation. I'm from Essex. Ever bags. All right, so you go first tickets to row there, So there's going to be in from once you infinity on export over in Victoria. Plus, So when n this zero, it is actually one and take Mr if there is going to be and from one to ou infinitely extra power on my sweat or in this one story Oh, and this is just we'll take if they would take care because the man's one So it's gonna be em from zero to in the expert over and over. And Victoria, which is just which is just equal to over afterthe X. So yet we have shown that we have ever that this function is solution to the first equation. I've probably cause f and B show that affect six either power fess. So the exponential function has our solution. All right, so we can just We can just Ah So this differential in clearance, it was the ex and we know that half of zero equals to one wasn't appear. So let's see here. So if we plug in zero on it becomes to one class extra power from one two videos. So it's off off in this zero eso yet We have this initial condition and the differential equation we can find with FX, so we're going to solve it. So this tell us, London Actually, Alex experts some constant and I think awesome. So the final answer is See a constant times a citizens, they want power. Congrats, ethnics. And we just felt in half of zero equals waas. So that implies, Say one equals one. And finally, I have dogs because into power X All right, so everything it's what Rex would have shown that.

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