Problem

Determine whether the series is convergent or div…

02:02

Problem 23 Easy Difficulty

Determine whether the series is convergent or divergent.
$ \displaystyle \sum_{k = 1}^{\infty} ke^{-k} $

Answer

convergent

Topics

Sequences

Series

Calculus: Early Transcendentals

Chapter 11

Infinite Sequences and Series

Section 3

The Integral Test and Estimates of Sums

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

Video Transcript

for this problem. We are going to use the ratio test and first I'm just going to simplify the ratio and then I'll take the limit. So we want to know is what does next term that is the K plus one term look like when it's divided by case turns. So I'm going to distribute on the numerator and split the fraction up so I can simplify things down. So I get Kay need the negative K minus one over K to the negative k. Let's eat the negative K minus one over. Okay, even negative K. Now the left hand side reduces to just mhm the negative one and the right hand side. It becomes each the negative one over K. Now, if we take the limit, this K goes to infinity of this we say that either the negative one over K goes to zero. And so we are left with just the two, the negative one and either the negative one is less than one. So the series is convergent by the ratio test

Clayton C.