Problem

Find the values of $ x $ for which the series con…

03:17

Question

Answered step-by-step

Problem 60 Medium Difficulty

Find the values of $ x $ for which the series converges. Find the sum of the series for those values of $ x. $
$ \displaystyle \sum_{n = 0}^{\infty} (-4)^n (x - 5)^n $

Video Answer

Solved by verified expert

Textbook Answer

Official textbook answer

Video by J Hardin

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Related Courses

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 11

Infinite Sequences and Series

Section 2

Series

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

Video Transcript

Let's go in and find the X values for which this Siri's converges that'LL be our first step. And then for those values of X that we get in part one. Well, go ahead and also find the sum of the Siri's so that'LL be our second step. So for the first step, the first thing I would do here is to rewrite this, since it's actually and geometric Siri's but kind of in disguise the way that it's written. So rewrite this by pulling out the N and the reason I could do that. I'm just using this fact here that you've seen before. You're laws of exponents. You could always do this. So I'm just using this fact with a equals minus four. B equals X minus five. So I see that this is geometric and I see that my our value, the thing that's being raised, the end power is negative four times X minus five. And then I know that a geometric series will converge, huh? If the absolute value bar is less than one, so in our case, we have absolute value, are so it's negative for parentheses, X minus five, and this all inside the absolute value. And so that's not a wonder that's absolute value of make out a little larger now. I could simplify this, said it less than one and then software X. So we have Let's divide by four and then using the definition of these absolute values here we have. And then at five all sides here five. Which you could raise twenty over four so have nineteen over for less than X, less than twenty one over four. So that's the answer for Part one, The's Air the Axis, for which the Siri's convergence now for Part two will assume that X is in this interval here. So that a convergence and now we'LL find the sum. So the sum for geometric series is always the first term of the Siri's. What you get by plugging in the starting number here and equal zero for N over one minus R. Now, in our problem, if we plug in and equal zero, we get zero exponents was So we just get a one in the numerator and then in the denominator we have one minus are and we know that our equals negative for Times X minus five. So watch out for that double negative there. So now and the denominator, we have one. And then what are we left over with? So how is one going to combine? So we have a plus here That's a plus for but then were multiplying. It's in negative five, so let's negative twenty. But we have a plus one here, so that's negative. Nineteen. But we also have positive four times X, so there's a for X there as well. So assuming that X is between nineteen over four and twenty one over, for the sum of the Siri's that was given to us is equal to one over for X minus nineteen, and that's your final answer.

Get More Help with this Textbook

James Stewart

Calculus: Early Transcendentals

View More Answers From This Book

Find Another Textbook

Study Groups

Study with other students and unlock Numerade solutions for free.

Problem