Geometric Series

The first thing that must be discussed when working with infinite series is the meaning of convergence of an infinite sequence of real numbers. A sequence is a function whose domain is the positive integers (sometimes the nonnegative integers). A sequence is denoted by $\{a_n\}$ where the $a_n$ refers to the $n$th term of the sequence. For example, if $\{a_n\}=\{1/n\}$, then the $100^{th}$ term of the sequence is $1/100$. A sequence converges to a real number $L$ if and only if only a finite number of terms of the sequence lie outside of any interval that has $L$ as its midpoint. Put another way, given any interval with $L$ as the midpoint, every term of the sequence after some term will lie in the interval. Every set of real numbers that is bounded has a least upper bound and a greatest lower bound. (This is known as the completeness property. The set of rational numbers does not possess this property.) A consequence of this property is that every bounded sequence that is nondecreasing (monotone increasing) or nonincreasing (monotone decreasing) converges. Which of the following sequences converge? $\{(.5)^n\}$, $\{n/(n+1)\}$, $\{1/n^2\}$, $\{(-1)^n\}$, $\{(-1)^n/n\}$.

An infinite series is formed by adding, successively, the terms of a sequence. If the sequence is $\{a_n\}=\{n\}$ then the series is $1 + 2 + 3 +\cdots$ which will be represented in sigma notation by $\sum_{k=1}^{n} k$. It is seen that this infinite series can exceed any given real number. On the other hand, if the sequence consists of terms that continue to get smaller and smaller, then it is not clear whether the sum will grow without bound.This is where the need for sequences comes in. For a given series, $\sum_{k=1}^{n} a_k$, form the sequence $\{S_n\}$, called the sequence of partial sums, where $S_1= a_1, S_2= a_1+a_2, S_3=a_1+a_2+a_3,$ and $S_n = a_1+a_2+a_3+ \cdots+a_n$. In words, $S_n$ is the sum of the first $n$ terms of the sequence $a_n$. After all of this build up we can now say what it means for a series to converge. A series converges if and only if the sequence of partial sums converges. The limit of the sequence of partial sums is said to be the sum of the series. Practice using this definition with some geometric series. Recall that a sequence is geometric if the ratio of any term and the preceding term is a constant. For example, $\{(-1)^n\}$, $\{(1)^n\}$,$\{(1/2)^n\}$, $\{(-1/2)^n\}$, and $\{(3/2)^n\}$ and $\{2002(1/2)^n\}$ are all geometric sequences. Which of the sequences converge? To decide which of the of the corresponding infinite series converge, we need to find the sequence of partial sums. Problem 1 will get you started.

1. Find $$\sum_{k=1}^{n} ar^{k-1} = a + ar + ar^2 + \cdots + ar^{n-1}$$.
2. Find $$\sum_{n=1}^{\infty} ar^{n-1} = a + ar + ar^2 + \cdots$$. For what values of $r$ does the series have a sum.?
3. What geometric series has a sum of $\frac{1}{1-x}$?
4. What geometric series has a sum of $\frac{4}{3+2x}$?
5. Find $$\sum_{k=1}^{2002} \frac{1}{2^k}$$.
6. Find $$\sum_{n=1}^{\infty} \frac{2001^n}{2002^n}$$.
7. Find $$\sum_{n=1}^{\infty} \frac{n}{2^n}$$.
8. (Mandelbrot Competition March 2002)
   Find $$\sum_{n=1}^{\infty} \frac{F_n}{3^n}$$, where $F_n$ is the $n$th Fibonacci number.
9. Find $$\sum_{n=1}^{\infty} \frac{5n+1}{3^n}$$.
10. Find $$\sum_{n=1}^{\infty} \frac{n^2}{4^k}$$.
11. Find $$\sum_{n=1}^{\infty} \frac{n^3}{3^n}$$. [answer 33/8]
12. Find $$\sum_{n=1}^{\infty} \frac{n^6}{2^n}$$. [answer 9366]