Converging Sequnces

1. Jun 5, 2012

trap101

Determine whether each of the following sequences converge or not. In each case present a formal
explanation. If a sequence converges find the limit and if not determine whether there should be any
converging subsequences, and if so find more than one converging subsequences.

Xk+1 = (k/k+2)Xk, where X1 = 1/2

Attempt: Now I was thinking of taking the (lim k-->∞ 1/(1+2/k) ) (lim k-->∞ Xk). In other words take the limits of the individual sequences and show that they converge, but I'm realizing I don't have an expression for Xk so I might not be able to do this. In the same breath if this sequence doesn't converge, would finding two subseqeunces through by fiddling with some numbers work?

2. Jun 5, 2012

hi trap101!

xk/x1 = … ?

3. Jun 5, 2012

Ray Vickson

As written, your sequence is xk+1 = 3xk, because k/k + 2 = 3. Is that what you were really given?

RGV

4. Jun 5, 2012

trap101

[k/(k+2)] Xk ......so the fraction k/(k+2) times Xk

.man I need to learn latex.......

5. Jun 6, 2012

Ray Vickson

No. Just use parentheses, as you did above.

Anyway, the easiest way to deal with the question is to find an actual formula for xk. If you write out in detail the first few values of xk you will see a pattern, and can then prove the general result by induction, for example.

RGV