Question

Does this sequence converge, if so, what's its limit?
An = 1.01
A_n+1 = 1/2(An^2 + 1) for n>=1 Does this sequence converge, if so, what's its limit?
An = 1.01
A_n+1 = 1/2(An^2 + 1) for n>=1 @Mathematics

Answer 1

\[a _{n+1}=1/2((a _{n})^2+1)\]

Answer 2

yes to 1

Answer 3

well the way you have it written it would be yes to 0

Answer 4

but im not really sure what you mean by an = 1.01

Answer 5

Oops sorry A1 = 1.01

Answer 6

how do we figure it out though

Answer 7

what happens as n gets really really big?

Answer 8

dont we need to prove tho that its bounded and increasing or decreasing tho?

Answer 9

our prof said we need to do that for recursive sequences

Answer 10

well its been a while since i did sequences but from what i remember you end up with a sequence of number like

a1, a2, a3, a4, a5, ... an

it converges if it exists and approaches some number

Answer 11

in your case as n gets really big you get 1/infinity

Answer 12

each number gets smaller and smaller until eventually it might as well be considered 0

Answer 13

sorry its 1/2* (....) the An^2 part is in the numerator, only the "2" is in the denom

Answer 14

in that case it would diverge off to infinity

Answer 15

you can tell this by looking at what happens as you plug in n's

Answer 16

the numbers get bigger and bigger

Answer 17

but we have an example that's not completed, and it said that a similar sequence converged, and it was 1/2*(An+6) with A1=2

Answer 18

even tho the first few terms were growing, it somehow converged

Answer 19

hold on a sec i cant quite remember sequences

Answer 20

ok

Answer 21

lim as n approaches infinity in the sequence (1/2)(An+6) converges off to infinity

Answer 22

i mean diverges off to infinity***

Answer 23

I dunno I have that the limit is 6 in my book, but they dont really go into how they found it

Answer 24

whats the question say exactly?

Answer 25

t asks "Does the sequence converge? If so, find it's limit, Otherwise, say it diverges" and it shows the recursive sequence above

Answer 26

i m 100 percent sure as you increase n in that situation the numbers get bigger than 6

Answer 27

they must be asking for something different.. are you sure they dont give a bound on n?

Answer 28

if n< or = 6 than it would converge to 6

Answer 29

well they give n >= 1

Answer 30

you or I must be missing something... not quite sure what. maybe one of the other guys in here could help ya i honestly dont know why that would = 6 and not infinity

Answer 31

=\ ouille this is a toughy

Answer 32

let me try

Answer 33

the sequence converges, if lim An+1/An = 1 , i believe

Answer 34

yes.. thats not what she gave tho

Answer 35

she said (1/2)(An+6) ; n>1

Answer 36

does the sequence converge, if so to what?

Answer 37

if An is bounded and monotonic, you can use monotonic sequence theorem

Answer 38

not sure what that is lol

Answer 39

im not sure how to use it tho.. what do i show it's bounded above/below to?

Answer 40

lets plug in some values to see where this is going

Answer 41

heres another idea

Answer 42

that thrm says that if An gets smaller and smaller than it has a limit

Answer 43

solve the algebraic equation

Answer 44

An is getting bigger

Answer 45

when i ddo that i get like
A1 = 1.01
A2 = 1.01005
A3 = 1.010100501
A4 = 1.010151511
A5 = 1.010203038

Answer 46

A_n+1 = 1/2A_n^2 +1/2, now subtract both sides A_n+1

Answer 47

ohh your looking at the top one

Answer 48

0 = 1/2 A_n^2 - A_n+1 + 1/2 , multiply both sides by 2 now

Answer 49

0 = A_n^2 - A_n+1 + 1

Answer 50

since for large n , A_n ~ A_n+1 , we can substitute and make a quadratic

Answer 51

err 0 = A_n^2 - 2A_n+1 + 1

Answer 52

so first substitute A_n for A_n+1

Answer 53

0 = A_n+1^2 - 2A_n+1 + 1 ,
0 = ( A_n+1 -1) ( A_n+1 -1)

Answer 54

A_n+1 = 1

Answer 55

it is easier to see this if you plug in A1 = 2, then it goes to 1 quickly

Answer 56

the limit is 1

Answer 57

wow i never knew you could do that trick where A_n ~ A_n+1 o_O

Answer 58

kirky, your values are wrong

Answer 59

A1 = 1.01
A2 = 1.01005
A3 = 1.010100501
A4 = 1.010151511
A5 = 1.010203038
this is false

Answer 60

A1 = 1.01
A2 = 1.005
A3 = 1.0025
A4 = 1.00125
A5= 1.000625

Answer 61

wait i made a mistake

Answer 62

wait, im wrong, this is divergent

Answer 63

i made a mistake, this is only true if |A1| < 1

Answer 64

oh

Answer 65

is there a way to prove its divergent

Answer 66

I think you can do a ratio test

Answer 67

We didn't get to the point in class where were at ratio tests =\... And.. isnt that for series only anyways? o.O

Answer 68

what i did only helps if we knew a priori that there was a limit

Answer 69

hmm.. ok well I dunno I can't seem to figure this one out but thanks for ur help tho

Answer 70

we can show that An+1 > An

Answer 71

ya i tried that and it worked, but i don't know how to show it's bounded.. since I'm not sure which bound I'm supposed to look for?

Answer 72

I don't know if it might be converging to a number or not o_o?

Answer 73

i dont think its converging

Answer 74

for instance try 1.5

Answer 75

as A1, that blows up fast

Answer 76

oh i see what u mean

Answer 77

even 1.1 blows up pretty fast

Answer 78

ok got it, it diverges

Answer 79

I am absolutely sure

Answer 80

because i sat there and hit that calculator key must be 100 times

Answer 81

starting 1.01

Answer 82

wow lol

Answer 83

ok i'll trust this then... hehe thanks again for your help :)!!! really appreciated