- #1
Saitama
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Homework Statement
Let ##x_k=k## for ##k \leq 31## and ##\displaystyle x_{k+1}=\frac{x_1+x_2+...x_k}{k}## for ##k \geq 31##. Also let ##y_k=x_k## for ##k \leq 31## and ##\displaystyle y_{k+1}=\frac{y_k+y_{k-1}+...y_{k-30}}{31}## for ##k \geq 31##. Now if ##z_k=y_k-x_k## for all ##k ε N##. Find ##\lim_{n→∞} z_n##.
Homework Equations
The Attempt at a Solution
I figured out that ##x_{k+1}=x_{k+2}=...=16##, so the question reduces to
[tex]\displaystyle \lim_{n→∞} z_n=y_n-16[/tex]
I am having trouble finding ##\lim_{n→∞}=y_n##.
I plugged in some numbers in the expression of ##y_k## starting k=31.
When k=31, ##y_{32}=16##.
When k=32, ##y_{33}=\frac{15+16*31}{31}##
The next terms go even more big. I am stuck here.
Any help is appreciated. Thanks!