Question

determine whether the sequence an=2e^(2n/(n+2)) converges or diverges. if it converges, find the limit

Answer 1

need help

Answer 2

Focus on 2n/(n+2) for a moment

Answer 3

what happens as n goes off to infinity?

Answer 4

alright

Answer 5

it increases?

Answer 6

if n = 1000, then what is 2n/(n+2) equal to?

Answer 7

it goes to infinty and beyond

Answer 8

its 1.996

Answer 9

a very small number...

Answer 10

@jim_thompson5910

Answer 11

ok now what is 2n/(n+2) equal to when n = 10,000 ?

Answer 12

an even smaller number

Answer 13

\[\frac{2n}{n+2} \approx \frac{2n}{n} \text{ for \large } n \]

Answer 14

1.999600000 to be exact

Answer 15

do you know where 2n/n goes as n continues to get bigger?

Answer 16

i dont know

Answer 17

ok just do it @jim_thompson5910 's way then

Answer 18

keep going, what about if n = 100,000 ?

Answer 19

do you want me to do the 2n/n or 2n/n+2?

Answer 20

let's use the original

Answer 21

i got 1.999960001

Answer 22

and if n = 1,000,000 then 2n/(n+2) = 1.999996000
and if n = 10,000,000 then 2n/(n+2) = 1.999999600
and if n = 100,000,000 then 2n/(n+2) = 1.999999960
and so on...

so if you keep bumping up n larger and larger, you'll find that 2n/(n+2) approaches 2

the leading terms 2n and n divide to get 2n/n = 2

Answer 23

oh alright

Answer 24

so back to 2e^(2n/(n+2))

Answer 25

what happens to 2e^(2n/(n+2)) as n --> infinity?

Answer 26

it approaches the number 15

Answer 27

what does 2e^(2n/(n+2)) simplify to when n approaches infinity?

give the exact form of it

Answer 28

u mean like use the L'hospital rule

Answer 29

let's say I had

sin(x+2)

and I wanted to know the exact value of that expression above when x = 4. I just plug in x = 4 and try to simplify as much as possible

sin(x+2) = sin(4+2) = sin(6)

sin(6) can't be simplified further so I leave it as it is

Answer 30

sin(6) has an approximate decimal form, but sometimes it's best to leave it in exact form

Answer 31

in this case we have 2e^2n/n+2 which simplifies to 2lne^2n/n+2 = 4n/n+2

Answer 32

no, we already know 2n/(n+2) approaches 2 as n approaches infinity

so just replace 2n/(n+2) with 2 and you're done

\[\LARGE \lim_{n \to \infty}\left(2e^{\frac{2n}{n+2}}\right) = 2e^{2}\]

Answer 33

so the whole thing approaches 2e^2 as n approaches infinity

Answer 34

so the sequence converges

Answer 35

yes because 2n/(n+2) converged

Answer 36

@freckles you still here?

Answer 37

@jim_thompson5910 i got a quick question

Answer 38

so you would do every exponent problem like this for example if we had 3 instead of 2 it would be 2e^3

Answer 39

?

Answer 40

what is the full problem

Answer 41

no just asking if we had another exponent problem like this one

Answer 42

yes you break it up like that

Answer 43

okay and what about when its a natural log problem you break it up too?

Answer 44

yes if you had ln( 2n/(n+2) ) you would focus on the 2n/(n+2) first and see that it approaches 2

so ln( 2n/(n+2) ) approaches ln( 2 )

Answer 45

okay thank you @jim_thompson5910

Answer 46

you're welcome