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
<br /> u_{k+1}+1&gt;3<br /> <br /> u_{k+1}&gt;2<br />

Thus u_{n+1}&gt;2 is true

<br /> (u_{k+1}+1)-3&lt;0<br /> <br /> u_{k+1}+1&lt;-3<br /> <br /> u_{k+1}&lt;-2<br />
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&lt;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<br /> <br /> <br /> <br /> <h2>Homework Equations</h2><br /> <br /> <h2>The Attempt at a Solution</h2><br /> u_{n+1}=-1+sqrt{u_n+7}<br /> <br /> \Rightarrow u_n=(u_{n+1}+1)^2-7<br /> <br /> Assume statement is true for all k\geq1<br /> <br />

<br /> <br /> Didn&#039;t you mean assume it&#039;s true for n\leq{k} ? Actually assuming it is true for n=k is sufficient. But either way is fine.<br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> then u_k&amp;lt;2<br /> \Rightarrow (u_{k+1}+1)^2-7&amp;lt;2<br /> <br /> <br /> (u_{k+1}+1)^2-(3)^2&amp;lt;0 </div> </div> </blockquote><br /> Good so far... but at this point I&#039;d take the 3^2 to the other side... if x^2&lt;a^2 where a&gt;0, then x&lt;a and x&gt;-a... and then you can use the x&lt;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...