Geometric Series: Finite Sums, Convergence, and Applications

All flashcards

What is the finite sum of a geometric series with initial term $a$, ratio $r\ne1$, and indices $0$ through $n$?

$S_n=\frac{a(1-r^{n+1})}{1-r}$.

What is the finite geometric sum when $r=1$?

Every term equals $a$, so $S_n=a(n+1)$.

What is the error after summing a convergent geometric series through index $n$?

$S-S_n=\frac{ar^{n+1}}{1-r}$.

How is a geometric series related to a power series?

It is a power series with the same coefficient at every power.

What is the present value of end-of-year payments $P$ made forever with discount factor $d$, where $0<d<1$?

$\frac{P}{1-d}$.

Quiz questions

1. For initial term $a$, ratio $r\ne1$, and terms indexed from $0$ through $n$, which formula gives the finite geometric sum?Answer: $S_n=\frac{a(1-r^{n+1})}{1-r}$
2. A finite geometric sum has initial term $a$, ratio $r=1$, and indices $0$ through $n$. What is its sum?Answer: $a(n+1)$
3. What is the sum of the infinite geometric series $\frac{3}{4}+\frac{3}{16}+\frac{3}{64}+\cdots$?Answer: $1$
4. For a convergent geometric series with initial term $a$ and ratio $r$, what is the signed difference $S-S_n$ after summing through index $n$?Answer: $\frac{ar^{n+1}}{1-r}$
5. A payment $P$ is made at the end of each year forever, with discount factor $d$ per year and $0<d<1$. What is its present value?Answer: $\frac{P}{1-d}$