Finding exact value of infinite series.

Question:

A tennis ball is dropped from $3$ meter drop and it bounces back upwards by $30\%$ of it's initial drop height. What is total distance traveled by tennis ball ?

Attempt to solve

Now we can measure distance traveled by tennis ball with sum defined as:

$$\sum_{i=0}^{\infty}3*0.3^{i}$$ where $i=$ number of bounces. Since we do not want to limit number of maximum bounces we set it to $\infty$.

plot of first 100 bounces. We can also see that our series is convergent.



Summation of 100 bounces will give us approximately:

$$\sum_{i=0}^{100}3*0.3^i \approx 4.285714286$$

There is a limit in what kind of total distance can be achieved which would be defined as:

$$\lim_{n\rightarrow \infty}\sum_{i=0}^{n}3*0.3^i=s$$

Now $s$ would be the value we want to know ? Now the problem is i don't know how to compute value for $s$.

• You have that $\sum_{i=0}^\infty q^i = 1/(1-q)$ for every $q\in (-1, 1)$. – Rigel Nov 4 '17 at 16:46

is a serie geometric So $$\sum_{i=0}^{\infty}3*0.3^{i}=3\sum_{i=0}^{\infty}0.3^{i}=3\cdot\frac{1}{1-0.3}$$ so is equal
$$\sum_{i=0}^{\infty}3*0.3^{i}=\frac{30}{7}\approx 4.28571428571$$
• How did you come up with $\frac{30}{7}$ ? – Tuki Nov 4 '17 at 17:02