Infinite Series Review

We will now review some of the recent content regarding infinite series.

• Recall from the Series page that a Series is the sum of all terms in a sequence, that is, if $\{ a_n \}$ is a sequence of real numbers then $\sum_{n=1}^{\infty} a_n$ is a series.

• For each $n = 1, 2, ...$ we define the $n^{\mathrm{th}}$ Partial Sum of the series to be $s_n = a_1 + a_2 + ... + a_n = \sum_{i=1}^{n} a_i$, and the sequence $\{ s_n \}$ is the corresponding Sequence of Partial Sums to the series $\sum_{n=1}^{\infty} a_n$.

• We said that a series $\sum_{n=1}^{\infty} a_n$ is Convergent if the sequence of partial sums $\{ s_n \}$ is convergent, and we said that the series is otherwise Divergent.

• We also noted that we can always rewrite a series by changing the start index when necessary. For example, if we have a series $\sum_{n=1}^{\infty} a_n$ and we wanted to start at $i = 3$ then we would could rewrite this series as:

(1)

\begin{align} \quad \sum_{n=1}^{\infty} a_n = \sum_{i=3}^{\infty} a_{i-2} \end{align}

• The first and perhaps easiest type of series that we looked at were Geometric Series. We defined a Geometric Series to be of the following form:

(2)

\begin{align} \quad \sum_{n=1}^{\infty} ar^{n-1} = a + ar + ar^2 + ... + ar^n + ... \end{align}

• The value $r$ is called the Common Ratio for the geometric series. We noted that the $n^{\mathrm{th}}$ partial sum of a geometric series is $s_n = \frac{a(1 - r^n)}{1 - r}$, and that geometric series converge to the sum $\frac{a}{1 - r}$ if and only if $\mid r \mid < 1$. If $\mid r \mid ≥ 1$ then the corresponding geometric series will converge. For example, the geometric series $\sum_{n=1}^{\infty} 2 \left ( \frac{1}{2} \right )^n$ converges to $\frac{2}{1 - \frac{1}{2}} = 4$, however, the geometric series $\sum_{n=1}^{\infty} 3 e^n$ diverges since the common ratio is $r = e$ and $\mid r \mid = \mid e \mid \approx 2.718... ≥ 1$.

• On The Harmonic Series page we looked at a very important series known as the Harmonic Series defined to be $\sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + ...$ which diverges.

• On the Telescoping Series page, we looked at Telescoping Series which are special series whose partial sums simplify to a fixed number of terms. For such series, we could then evaluate the limit of the partial sum of the series to find the sum of the series. For example, consider the following series:

(3)

\begin{align} \quad \sum_{n=2}^{\infty} \frac{1}{n^2 - n} = \sum_{n=2}^{\infty} \left ( \frac{1}{n - 1} - \frac{1}{n} \right ) = \left ( 1 - \frac{1}{2} \right ) + \left ( \frac{1}{2} - \frac{1}{3} \right ) + ... + \left ( \frac{1}{n -1} - \frac{1}{n} \right ) + ... \end{align}

• Notice that the sequence of partial sums $s_n = 1 - \frac{1}{n}$ and $\lim_{n \to \infty} \left ( 1 - \frac{1}{n} \right ) = 1$, so the series above converges to $1$.

• We then looked more into the Properties of Convergent Series. The properties of convergent series follow immediately from the properties of convergent sequences and are summarized in the table below.

Sum Law of Series	If $\sum_{n=1}^{\infty} a_n = A$ and $\sum_{n=1}^{\infty} b_n = B$ then $\sum_{n=1}^{\infty} (a_n + b_n) = A + B$.
Difference Law of Series	If $\sum_{n=1}^{\infty} a_n = A$ and $\sum_{n=1}^{\infty} b_n = B$ then $\sum_{n=1}^{\infty} (a_n - b_n) = A - B$.
Product Law of Series	If $\sum_{n=1}^{\infty} a_n = A$ and $\sum_{n=1}^{\infty} b_n = B$ then $\sum_{n=1}^{\infty} (a_nb_n) = AB$.
Comparison Law	If $a_n ≤ b_n$ for all $n \in \mathbb{N}$, $\sum_{n=1}^{\infty} a_n = A$ and $\sum_{n=1}^{\infty} b_n = B$ then $A ≤ B$.

• On the Convergence and Divergence Theorems for Series page, we also looked at some important theorems The most important theorem on this page said that $\sum_{n=1}^{\infty} a_n$ converges if and only if for each $N \in \mathbb{N}$ the series $\sum_{n=N}^{\infty} a_n$ converges. This theorem implies that the convergence of a series depends only on the tail of the series.