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## Homework Statement

Let [tex]U_{n}=\frac{n^{2}}{2^{n}}[/tex] for every n in N

1) For every n>0 let [tex]V_{n}=\frac{U_{n+1}}{U_{n}}[/tex]

a) Prove that [tex]\lim V_{n}=\frac{1}{2}[/tex]

b) For every n>0 prove that: [tex]V_{n}>\frac{1}{2}[/tex]

c) First the smallest natural number N such that : [tex]n\geq N\Rightarrow V_{n}<\frac{3}{4}[/tex]

d) Conclude that [tex]n\geq N\Rightarrow U_{n+1}<\frac{3}{4}U_{n}[/tex]

2) We want to show that [tex](S_{n})_{n\geq5}[/tex] is convergent such that:

Sn=U5+U6+U7+....+Un

a) Prove by induction that for every natural number greater than 5: [tex]U_{n}<(\frac{3}{4})^{n-5}U_{5}[/tex]

b) Prove also by induction that for every natural number greater than 5:

Sn≤[1+(3/4)+(3/2)^2+....+(3/4)^(n-5)]U5

c) Conclude that Sn≤4U5 for every n≥5

3) Prove that [tex](S_{n})_{n\geq5}[/tex] is monotone increasing and conclude that it is convergent.

## The Attempt at a Solution

Solved 1) a and b and stuck on c and d.

For number 2-a I showed that U5≤U5 and I need to know how to show that Un+1≤(3/4)^n-4U5.

I have no idea on b and c and number 3.

Thanks for any help before hand.