Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Use the Ratio Test to determine whether the series is convergent or divergent.$ \displaystyle \sum_{n = 1}^{\infty} \frac {(2n)!}{(n!)^2} $

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

divergent

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 6

Absolute Convergence and the Ratio and Root Tests

Sequences

Series

Missouri State University

Campbell University

University of Michigan - Ann Arbor

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.

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.

03:44

Use the Ratio Test to dete…

03:31

03:41

05:19

Use the ratio test to deci…

0:00

02:56

accused the ratio test to determine whether the Siri's convergence. So let's look at this term here and Jeff two in Factorial, Divided by and Factorial Square. Now the ratios has suggests that we look at the limit and goes to infinity absolute value and plus one over. And let me go back there. Live a little sloppy a n plus one over an. Now we can drop absolute value here because AM is positive. So the numerator let me do this in red. This is too, and plus two so noticed that if you increase enter and plus one, this is two, one plus two and then we have factorial and then n plus one factorial and that's clear divided by hand. Now let's go ahead and simplify this a little bit. So this term up here, this to one plus two factorial I can write that is two in factorial two in factorial times two in plus one times two, one plus two. That's this term right here. And then I could also bring this and Fats for real square up into the numerator. And then I'm left over with here I penis two in factorial and then I have n plus one Factorial saw me, right? That is and factorial times and plus one and then that square. So this is corresponding to this term. Over here correspond Silvestre. And of course, here I should write that limit on the front somewhere living is and goes to infinity. Let me bring this down to the bottom left. So here we should cancel out as much as we can. We see that we can take off those to inventory ALS here we have in fact, Foreal Square that'LL cancel off with this and factorial that's being squared. And then if we walk around, we see that in the numerator we still have two in plus one two in plus two In the denominator, you have n plus one square. This limit is equal to four. Just look at the leading terms up these quadratic CE. This is bigger than one. So we conclude that the Siri's from one to infinity two in factorial over. In fact, for the square damages one by the ratio test. And that's our unless, sir

View More Answers From This Book

Find Another Textbook

Numerade Educator

Oregon State University

Baylor University

03:00

"please solve all of this please don't spam 2 Find the SUm of …

02:06

'please help .......If y97 97 is divided by y + 1, the remainder is…

02:23

'solve this question solve this questionQNO-(8): Find the value of…

00:34

'pls give the write ans step by step explanation i will thank your 10 a…

03:08

'In the given figure, PA, QB and RC are each perpendicular to AC. If AP…

'help plzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzWrite_Tfor tru…

13:13

'(i) the total surface area,of the solid cylinder with conical ends as …

00:58

'5. Estimate the following(i) 243 + 4272(v) 243 + 1252(i) 243 + 427…

00:35

'maths quiz solve properly pls give me right answer4) Identify the …

01:14

'Find the answer please In AXYZ and ATUV, XY= 5 cm; YZ = 6 cm &…