# Thread: A couple Calc 2 questions

1. ## A couple Calc 2 questions

1) There is an infinite staircase. Find the total volume of the staircase, given that the largest cube has a side of length 1 and each successive cube has a side whose length is half that of the preceding cube.

2) Show that for all real values of x
sinx-(1/2sin^(2)x)+(1/4sin^(3)x)-(1/8sin^(4)x)+...=(2sinx)/(2+sinx)

2. Hello, noles2188!

These are geometric series.

The infinite sum is: .$\displaystyle S \:=\:\frac{a}{1-r}\quad\text{where: }\:\begin{Bmatrix}a \:=\:\text{first term} \\ r \:=\:\text{common ratio} \end{Bmatrix}$

1) There is an infinite staircase.
Find the total volume of the staircase, given that the largest cube has a side of length 1
and each successive cube has a side whose length is half that of the preceding cube.

We have: .$\displaystyle V \;=\;1^3 + \left(\frac{1}{2}\right)^3 + \left(\frac{1}{4}\right)^3 + \left(\frac{1}{8}\right)^3 + \left(\frac{1}{16}\right)^3 + \hdots$

. . . . . . . .$\displaystyle V \;=\;1 + \frac{1}{2^3} + \frac{1}{2^6} + \frac{1}{2^9} + \frac{1}{2^{12}} + \hdots$

This is a geometric series with: .$\displaystyle a = 1,\;\;r = \frac{1}{2^3} = \frac{1}{8}$

Its sum is: .$\displaystyle V \;=\;\frac{1}{1-\frac{1}{8}} \;=\;\frac{1}{\frac{7}{8}} \;=\; \frac{8}{7}$

2) Show that for all real values of $\displaystyle x\!:\;\;\sin x-\tfrac{1}{2}\sin^2\!x + \tfrac{1}{4}\sin^3\!x - \tfrac{1}{8}\sin^4\!x +\hdots \:=\:\frac{2\sin x}{2 + \sin x}$

This is a geomtric series with: .$\displaystyle a = \sin x,\;\;r = -\tfrac{1}{2}\sin x$

Its sum is: .$\displaystyle \frac{\sin x}{1 - \left(\text{-}\frac{1}{2}\sin x\right)} \;=\;\frac{\sin x}{1 + \frac{1}{2}\sin x}\cdot{\color{blue}\frac{2}{2}} \;=\;\frac{2\sin x}{2 + \sin x}$