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(a) Show that $ \sum_{n=0}^{\infty} x^n/n! $ converges for all $ x. $

(b) Deduce that $ lim_{n \to \infty} x^n/n! = 0 $ for all $ x. $

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a. $\sum_{n=0}^{\infty} \frac{x^{n}}{n !}$b. the series of part (a) always converges, we must have $\lim _{n \rightarrow \infty} \frac{x^{n}}{n !}=0$ by Theorem 11.2 .6

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 6

Absolute Convergence and the Ratio and Root Tests

Sequences

Series

Campbell University

Harvey Mudd College

Baylor University

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.

01:57

(a) Show that $\sum_{n=0}^…

05:35

(a) Show that $ \sum_{n=0}…

01:32

(a) Show that $\Sigma_{n-0…

03:13

Show that $$\sum_{n=1}…

Let's show that this Siri's converges for all ex. Some party, eh? With use the ratio test for this is our r A n So the ratio test for requires us to look at a N plus one over a end absolute value of that and then take the limit is and goes to infinity. So here will have so that a m plus one of the top and then they end in the bottom. Now we should go ahead and flip that red fraction over and multiply dividing by a fraction here. So we'LL have X and plus one over X n and then in factorial over and plus one that's for real. So here's the key observation here. You just look at this fraction. So using the definition of the factorial function, she write that out, You could see that we could cancel the first and batters and you just slept over with one over and plus one. So here will have just one ex leftover eggs on top and plus one in the bottom. But here, for any real Number X that we have excessive fixed number, it's not a limit here. The end is in the limit, and this will always go to zero due to that denominator. So you could think of this is being X over infinity. But X is not infinity or minus Infinity X is a real number, so we have zero, which is less than one. So we conclude that it converges for all ex, and that was the majority of the work here because part B will follow from the test for divergence. So this is a corollary of the test for diversions. So sorry here, mixing on my letters. The test for diversion says if the Siri's converges, then the limit of the and has to go to zero. So that's just the theorem in the book. But in our case, the limit of a end based on our definition of an this is just a limit as n goes to infinity X to the end over and factorial. So you go back and erase that. That's and factorial there kind of running out of room here. Scylla meeting There's my in factorial and then by the serum above that were citing, This has to be zero and that our deduction in part B and that completes our answer

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