# Series/Convegence tests! need help on review problems for test!

• Oct 29th 2007, 09:34 PM
xfyz
Series/Convegence tests! need help on review problems for test!
1. A certain ball has the property that each time it falls from a height h onto a hard, level surface, it rebounds to a height rh, where 0 < r < 1. Suppose that the ball is dropped from an initial height of H meters.
a) Assuming the ball bounces indefinitely, find the total distance that it travels (Use the fact that the ball falls 1/2gt^2 meters in t seconds)
b)calculate the total time that the ball travels

2. A series is Sum an is defined by the equations:

a1 = 1 an+1 = [(2+cosn)/(sqrt(n)]*an

Determine whether Sum an converges or diverges.

could someone show me how to do these? going NUTS.
thank u
• Oct 30th 2007, 06:35 AM
Soroban
Hello, xfyz!

I'll get you started on #1 . . .

Quote:

1. A certain ball has the property that each time it falls from a height $\displaystyle h$
onto a hard, level surface, it rebounds to a height $\displaystyle rh$, where $\displaystyle 0 < r < 1.$
Suppose that the ball is dropped from an initial height of $\displaystyle H$ meters.

a) Assuming the ball bounces indefinitely, find the total distance that it travels.

Let's baby-talk our way through the first few bounces . . .

First, it falls $\displaystyle H$ meters.

Then it bounces up $\displaystyle Hr$
m . . . and falls $\displaystyle Hr$ m.

Then it bounces up $\displaystyle (Hr)r = Hr^2$ . . . and falls $\displaystyle Hr^2$

Then it bounces up $\displaystyle (Hr^2)r = Hr^3$ . . . and falls $\displaystyle Hr^3$

. . and so on . . .

The total distance is: .$\displaystyle D \;=\;H + 2Hr + 2Hr^2 + 2Hr^3 + \cdots$

. . $\displaystyle \text{and we have: }\;D \;=\;H + 2Hr\underbrace{\left(1 + r + r^2 + r^3 + \cdots\right)}_{\text{geometric series}}$

The series in paretheses has first term 1 and common ratio $\displaystyle r.$
. . Its sum is: .$\displaystyle \frac{1}{1-r}$

So we have: .$\displaystyle D \;=\;H + 2Hr\left(\frac{1}{1-r}\right)\quad\Rightarrow\quad\boxed{D \;=\;H\left(\frac{1+r}{1-r}\right)}$

• Oct 30th 2007, 07:09 AM
ThePerfectHacker
Quote:

Originally Posted by xfyz
2. A series is Sum an is defined by the equations:

a1 = 1 an+1 = [(2+cosn)/(sqrt(n)]*an

Determine whether Sum an converges or diverges.

could someone show me how to do these? going NUTS.
thank u

I think is converges.

First note that,
$\displaystyle a_{n+1} = \left( \frac{2+\cos n}{\sqrt{n}} \right) \left( \frac{2+\cos (n-1)}{\sqrt{n-1}} \right) ... \left( \frac{2+\cos 1}{\sqrt{1}}\right)$.
Thus,
$\displaystyle |a_{n+1}| = \left| \frac{2+\cos n}{\sqrt{n}}\right| \cdot ... \cdot \left| \frac{2+\cos 1}{\sqrt{1}} \right| \leq \frac{3}{\sqrt{n}}\cdot ...\cdot \frac{3}{\sqrt{1}} = \frac{3^n}{\sqrt{n!}}$
Use the root test,
$\displaystyle \left| \frac{3^n}{\sqrt{n!}}\right|^{1/n} = \frac{3}{\sqrt{(n!)^{1/n}}} \to 0$ because $\displaystyle (n!)^{1/n} \to \infty$.
Thus, by the comparison test this series converges.