Welcome to Section 11.2: Infinite Series!
In this lecture, we'll be diving into the fascinating world of infinite series. We will explore what it means for an infinite series to converge or diverge and introduce some essential tools for determining their behavior. Let's embark on this journey together!
What is an Infinite Series?
An infinite series is essentially the sum of an infinite sequence. If we have a sequence ${a_n}_{n=1}^{\infty}$, then the infinite series formed by this sequence is:
$$\sum_{n=1}^{\infty} a_n = a_1 + a_2 + a_3 + \dots$$
Think of it as adding up infinitely many terms! But does this sum always make sense? That's where the concepts of convergence and divergence come into play.
Convergence and Divergence
To determine if an infinite series converges, we look at its partial sums. The $n$th partial sum, denoted by $s_n$, is the sum of the first $n$ terms of the series:
$$s_n = \sum_{i=1}^{n} a_i = a_1 + a_2 + \dots + a_n$$
If the sequence of partial sums ${s_n}$ converges to a real number $s$ as $n$ approaches infinity (i.e., $\lim_{n \to \infty} s_n = s$), then we say that the infinite series converges, and its sum is $s$. We write:
$$\sum_{n=1}^{\infty} a_n = s$$
Otherwise, if the sequence of partial sums does not converge, the series is said to diverge.
Example 1: Finding the Sum from Partial Sums
Suppose we know that the sum of the first $n$ terms of a series is given by:
$$s_n = \frac{2n}{3n + 5}$$
To find the sum of the infinite series, we take the limit as $n$ approaches infinity:
$$\lim_{n \to \infty} s_n = \lim_{n \to \infty} \frac{2n}{3n + 5} = \frac{2}{3}$$
Therefore, the series converges, and its sum is $\frac{2}{3}$.
Example 2: Convergent Series with Partial Fractions
Let's investigate the series $\sum_{n=1}^{\infty} \frac{1}{n(n+1)}$. To determine its convergence and find its sum, we can use partial fraction decomposition:
$$\frac{1}{n(n+1)} = \frac{A}{n} + \frac{B}{n+1}$$
Solving for $A$ and $B$, we find $A = 1$ and $B = -1$. Thus, we can rewrite the series as:
$$\sum_{n=1}^{\infty} \left(\frac{1}{n} - \frac{1}{n+1}\right)$$
The partial sums telescope, and we find:
$$\sum_{n=1}^{\infty} \left(\frac{1}{n} - \frac{1}{n+1}\right) = 1$$
Hence, the series converges to 1.
Geometric Series: A Special Case
A geometric series has the form:
$$\sum_{n=1}^{\infty} ar^{n-1} = a + ar + ar^2 + ar^3 + \dots$$
where $a$ is the first term and $r$ is the common ratio. A geometric series converges if $|r| < 1$, and its sum is given by:
$$\sum_{n=1}^{\infty} ar^{n-1} = \frac{a}{1 - r}$$
If $|r| \geq 1$, the geometric series diverges.
Test for Divergence
An important theorem states: If the series $\sum_{n=1}^{\infty} a_n$ is convergent, then $\lim_{n \to \infty} a_n = 0$.
This leads to the Test for Divergence: If $\lim_{n \to \infty} a_n$ does not exist or if $\lim_{n \to \infty} a_n \neq 0$, then the series $\sum_{n=1}^{\infty} a_n$ is divergent.
Properties of Convergent Series
If $\sum a_n$ and $\sum b_n$ are convergent series, then the following hold:
- $\sum ca_n = c \sum a_n$ (where $c$ is a constant)
- $\sum (a_n + b_n) = \sum a_n + \sum b_n$
- $\sum (a_n - b_n) = \sum a_n - \sum b_n$
Keep practicing, and you'll master these concepts in no time!