Converging Seq: Determine Limit of Xk+1 = (k/k+2)Xk, X1=1/2

[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.

X_k+1 = (k/k+2)X_k, where X₁ = 1/2


Attempt: Now I was thinking of taking the (lim k-->∞ 1/(1+2/k) ) (lim k-->∞ X_k). 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 X_k 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?

 

[tiny-tim]

hi trap101!

x_k/x₁ = … ? ​

 

[Ray Vickson]

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.

X_k+1 = (k/k+2)X_k, where X₁ = 1/2


Attempt: Now I was thinking of taking the (lim k-->∞ 1/(1+2/k) ) (lim k-->∞ X_k). 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 X_k 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?

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

RGV

 

[trap101]

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

RGV

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

.man I need to learn latex...

 

[Ray Vickson]

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

.man I need to learn latex...

No. Just use parentheses, as you did above.

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

RGV