Question: I have some troubles wit the next exercise.

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> Find the required work for move a particle from the point $(1,0)$ to the point $(-1,0)$ trought the ellipse $x^2+\displaystyle\frac{y^2}{b^2}=1$ against to the force $F(x,y)=(3y^2+2,16x)$. Which value of $b$ minimizes the work?
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First, consider $\gamma(t)=(\cos(t),-b\sin(t))$ the parametrization of the ellipse. The sign in the second coordinate is because the field goes counterclockwise, and, then, the parametrization should be goes clockwise.

Then, $$\displaystyle\int\_{0}^{\pi}(3b^2\sin^2(t)+2,16\cos(t))\cdot(-\sin(t),-b\cos(t))\,dt$$The integral then is $$\int\_{0}^{\pi}-3b^2\sin^3(t)-2\sin(t)-16b\cos^2(t)\,dt$$And the value is $-4b^2-8b\pi-4$ but, this function of $b$ doesn't have minimum, only a maximum. What happened here? What is wrong with my exercise?

Answer: 
Let $a\_0=0$ and $a\_{n+1}=\cfrac1{1+\cfrac1{5+a\_n}}$.

Prove by induction:

1) $a\_n<\dfrac{-5+\sqrt{45}}2$

2) $a\_{n+1}>a\_n$ i.e. monotone

So that it converges and that it must converge to some $a'$ such that:

$$a'=\cfrac1{1+\cfrac1{5+a'}}$$

(feel free to ask if you need more tips on any steps below, hover on the below tips to see major steps)

Induction step on proving 1)

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>  \begin{align}&a\_0<\cfrac{-5+\sqrt{45}}2\\&a\_{n+1}=\cfrac1{1+\cfrac1{5+a\_n}}<\cfrac1{1+\cfrac1{5+\cfrac{-5+\sqrt{45}}2}}=\cfrac{-5+\sqrt{45}}2\end{align}
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Induction on 2)

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>  \begin{align}&a\_1>a\_0\\&a\_{n+1}=\cfrac1{1+\cfrac1{5+a\_n}}>\cfrac1{1+\cfrac1{5+a\_{n-1}}}=a\_n\end{align}
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Hint : Simply consider the difference between two successive recurring terms and then it is easy to show that the sequence is monotonically increasing.  

 Now, $$\frac{1}{5} > 0\\\implies1+ \frac{1}{5} > 1\\\frac{1}{1 +\frac{1}{5}} < 1$$. 

Use this argument several times and try to deduce that the given sequence is bounded above by 1 and then by monotone-bounded theorem, you know that there exists a limit of the sequence. Let it be l. Now a routine computation yields your answer.

N.B.: in order to show that the sequence is monotone , I guess you will need to show by induction that $$a\_n < \frac{-5 + \sqrt{45}}{2}$$. And this can also come handy to show that the sequence is bounded above.