Problem

Determine whether the sequence converges or diver…

05:35

Problem 55 Hard Difficulty

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = \frac {n!}{2^n} $

Answer

diverges.

Topics

Sequences

Series

Calculus: Early Transcendentals

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Discussion

You must be signed in to discuss.

Top Calculus 2 / BC Educators

Watch More Solved Questions in Chapter 11

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

Problem 31

Problem 32

Problem 33

Problem 34

Problem 35

Problem 36

Problem 37

Problem 38

Problem 39

Problem 40

Problem 41

Problem 42

Problem 43

Problem 44

Problem 45

Problem 46

Problem 47

Problem 48

Problem 49

Problem 50

Problem 51

Problem 52

Problem 53

Problem 54

Problem 55

Problem 56

Problem 57

Problem 58

Problem 59

Problem 60

Problem 61

Problem 62

Problem 63

Problem 64

Problem 65

Problem 66

Problem 67

Problem 68

Problem 69

Problem 70

Problem 71

Problem 72

Problem 73

Problem 74

Problem 75

Problem 76

Problem 77

Problem 78

Problem 79

Problem 80

Problem 81

Problem 82

Problem 83

Problem 84

Problem 85

Problem 86

Problem 87

Problem 88

Problem 89

Problem 90

Problem 91

Problem 92

Problem 93

Video Transcript

for this problem. Recall that in factorial is just one times two times three times. Stop that thought all the way up to times in and to the end is just two times two times two times two where there's in total copies of two there. So if we rewrite this, it's one half times two over too. Times three over, too. Not got times in over two. So if we get some bounds on this, we know that this is going to be greater than one half times to over two times three over to and all of the terms over here, all those terms, they're going to be bigger than three over to. So if we just replace all of them with three over to, then we're going to get something smaller, okay? And there's a total of in minus two terms over there. Right, Because we have it. We started with in terms total, and then we have this one half term, and we have this to over two term. So after taking those out, we have in minus two terms left. Okay, so we get this bound happening here, and what that means is that if n goes to infinity. That's going to be greater than limit as N goes to infinity of one half times, two over to two or two is just one times three over, too, to the end, minus two three over two is bigger than one so through over to is bigger than one. And that tells us that this term, through over to to the in minus two power is going to blow up to infinity. So certainly multiplying by one half, it's still going to be infinity. So if our AI in terms has in goes to infinity, are bigger than infinity than they have to blow up a swell right. So we also have to have that limit. As n goes to infinity of Anne, that also has blow up in his infinity, so our sequence diverges.

Gabriel R.

Topics

Sequences

Series

Calculus: Early Transcendentals

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Problem