Geometric sums

For each of the following infinite geometric series, determine the values of $a$ and $r$, compute the partial sums $S_{5}$ and $S_{10}$ exactly (writing each as a fraction), and if the infinite geometric series converges, find its value.

1. $1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \cdots$

2. $4 - 2 + 1 - \frac{1}{2} + \frac{1}{4} - \cdots$

3. $2 + \frac{8}{3} + \frac{32}{9} + \frac{128}{27} + \cdots$

4. $\sum_{k = 0}^{\infty} 5 \cdot \left( \frac{3}{4} \right)^k$

5. $\sum_{k = 1}^{\infty} -2 \cdot \left( -\frac{2}{3} \right)^k$

Consider the function $f(x) = \frac{1}{1-x}$. Find the degree $n$ Taylor polynomial approximation centered at $a = 0$ for $f$.

1. To begin finding $T_n(x)$, do the usual work of computing the various derivatives of $f$ and their respective values at $a = 0$; note that it's helpful to view $f(x)$ in the form $f(x) = (1-x)^{-1}$ so that we can easily compute the derivatives of $f$ using the chain rule. For instance, where the first “$-1$” arises from the power rule, while th…

Suppose you drop a golf ball onto a hard surface from a height $h$. The collision with the ground causes the ball to lose energy and so it will not bounce back to its original height. The ball will then fall again to the ground, bounce back up, and continue. Assume that at each bounce the ball rises back to a height $\frac{3}{4}$ of the height from which it dropped. Let $h_n$ be the height of the ball on the $n$th bo…