7.5 Infinite geometric series

Evaluate the following series. If your answer is divergent, write “None”

(a) [math]1-0.2+0.04-0.008+\cdots[/math]

(b) [math]3-3+3-3+3-3+\cdots[/math]

If [math]1+\displaystyle{\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{6}}+\frac{1}{\sqrt{18}}+\frac{1}{\sqrt{54}}}+\cdots=a+b\sqrt{2}+c\sqrt{6}[/math] where [math]a, b[/math], and [math]c[/math] are rational numbers, what is the value of [math]a+b+c[/math]?

Rewrite the following numbers in fraction form.

(a) [math]0.3\overline{25}[/math]

(b) [math]6.\overline{099}[/math]

(c) [math]3.9\overline{2}[/math]

The value of [math]\dfrac{1}{11}+\dfrac{1}{101}+\dfrac{1}{1001}+\dfrac{1}{10001}+\cdots[/math] is

Let [math]\{a_n\}[/math] be a sequence satisfying [math]a_n=\left(\dfrac{\sqrt{5}+1}{4}\right)^n+\left(\dfrac{\sqrt{5}-1}{4}\right)^n[/math]. If[math]\displaystyle{\sum^\infty_{k=1}a_k}=a+b\sqrt{5}[/math] where [math]a[/math] and [math]b[/math] are rational numbers, what are [math]a[/math] and [math]b[/math]?

Consider a unit square [math]E_1[/math]. We construct another square [math]E_2[/math] whose vertices are the midpoints of [math]E_1[/math], then we use the same method to construct [math]E_3, E_4, \cdots[/math].

(a) Find the area of the shaded region.

(b) Find [math]\displaystyle{\sum^\infty_{k=0}area(E_k)}[/math]

Observe the figure above. Assume that the greatest circle is a unit circle. Find the summation of the area of all circles.

A tennis ball that is released to bounce freely reaches 30 inches on the first bounce. On each successive bounce, the tennis ball only reaches 75% of the height of the previous bounce. What is the total distance traveled by the tennis ball before it stops bouncing? (Note: each bounce is counted as a round trip: up and down)