Help in proving sequence by induction

[rock.freak667]

Homework Statement

The sequence $$u_1,u_2,u_3,...$$ is such that $$u_1=1$$ and $$u_{n+1}=-1+{\sqrt{u_n+7}}$$

a) Prove by induction that $$u_n<2 for all n\geq1$$
b) show that if $$u_n=2-\epsilon$$, where $$\epsilon$$ is small, then $$u_{n+1}\approx 2-\frac{1}{6}\epsilon$$

Homework Equations

The Attempt at a Solution

$$u_{n+1}=-1+sqrt{u_n+7}$$

$$\Rightarrow u_n=(u_{n+1}+1)^2-7$$

Assume statement is true for all $$k\geq1$$
then $$u_k<2$$
$$\Rightarrow (u_{k+1}+1)^2-7<2$$


$$(u_{k+1}+1)^2-(3)^2<0$$

$$((u_{k+1}+1)-3)((u_{k+1}+1)-3)<0$$

$$(u_{k+1}+1)-3>0$$ AND $$(u_{k+1}+1)-3<0$$
$$u_{k+1}+1>3 u_{k+1}>2$$

Thus $$u_{n+1}>2$$ is true

$$(u_{k+1}+1)-3<0 u_{k+1}+1<-3 u_{k+1}<-2$$
does this affect anything in my proof?

I didn't bother to substitute the values of $$u_1$$ and $$u_2$$ and so forth as i have already done it and it is so for all $$n\geq1$$

but I do not know how to do part b)

 

[learningphysics]

Homework Statement

The sequence $$u_1,u_2,u_3,...$$ is such that $$u_1=1$$ and $$u_{n+1}=-1+{\sqrt{u_n+7}}$$

a) Prove by induction that $$u_n<2 for all n\geq1$$
b) show that if $$u_n=2-\epsilon$$, where $$\epsilon$$ is small, then [tex]u_{n+1}\approx 2-\frac{1}{6}\epsilon

Homework Equations

The Attempt at a Solution

$$u_{n+1}=-1+sqrt{u_n+7}$$

$$\Rightarrow u_n=(u_{n+1}+1)^2-7$$

Assume statement is true for all $$k\geq1$$

Didn't you mean assume it's true for $$n\leq{k}$$ ? Actually assuming it is true for n=k is sufficient. But either way is fine.

then $$u_k<2$$
$$\Rightarrow (u_{k+1}+1)^2-7<2$$


$$(u_{k+1}+1)^2-(3)^2<0$$

Good so far... but at this point I'd take the 3^2 to the other side... if x^2<a^2 where a>0, then x<a and x>-a... and then you can use the x<a part to finish...

 

[learningphysics]

For part b, use the taylor series expansion in terms of $$\epsilon$$... you only need the first two terms.

 

[rock.freak667]

So then expand $$\sqrt{u_n+7}$$ using taylor series and it should work out then?

 

[learningphysics]

So then expand $$\sqrt{u_n+7}$$ using taylor series and it should work out then?

You should substitute in $$2-\epsilon$$ for $$u_n$$ in the expression for $$u_{n+1}$$... then get the taylor series of $$u_{n+1}$$ in terms of epsilon...