Question

Suppose a sequence is defined as a1=1 and a_n+1= (3*a_n)/(9+a_n). Assuming {a_n} is convergent, what is the limit of the sequence????

Answer 1

Damn recursive sequences...

Answer 2

Finding limits, I assume he/she already knows about sequences.

Answer 3

yup, thats the theory of recursive sequences. I just wanted help finding the limit as n goes to infinity.

Answer 4

Why not write out the first five or so terms of the sequence, starting with a1=1?
The pattern should become evident.

Answer 5

ya i made an excel spreadsheet and found that it pretty much goes to 0, but I wanted to find a more rigorous solution.

Answer 6

\[a _{1}=1;a _{2}=\frac{ 3(1) }{ 9+1 }=\frac{ 3 }{ 10 }; a _{3}=\frac{ 3(?) }{ 9+(?) }=?\]

Answer 7

as n->infinity, a_n+1=a_n, so assume that is x, u get the eqn x=3x/(9+x)...

Answer 8

Dave, not to challenge your comment, but upon what do you base your statement that a(n+1)=a(n) as n increases without bound?

Answer 9

so rearranging u get x(x+6)=0 but x=-6 is not a solution as all a_n are +, a_n=0 as n->infinity

Answer 10

because it is given "Assuming {a_n} is convergent"

Answer 11

So, Dave, because in the end a(n) approaches some fixed value, we dub a(n) "convergent"? I do think that's reasonable.

Answer 12

converging means....nevermind ur description is good @mathmale :)

Answer 13

lol @ mathmale

Answer 14

thanks dave

Answer 15

welcome