Series

Series and Partial Sums

Definition: If $\{ a_n \}_{n=1}^{\infty}$ is an infinite sequence, then we say that the sum of all terms in the sequence denoted $\sum a_n = \sum_{n=1}^{\infty} a_n = a_1 + a_2 + ...$ is the Infinite Series of the sequence $\{ a_n \}$. Furthermore, we say that $s_m = \sum_{n=1}^{m} a_n = a_1 + a_2 + ... + a_{m-1} + a_m$ to be the $m^{\mathrm{th}}$ Partial Sum of $\{ a_n \}$.

For example, consider the sequence $\{ n \}_{n = 1}^{\infty}$. The corresponding infinite series to this sequence would be:

(1)

\begin{align} \sum_{n=1}^{\infty} a_n = 1 + 2 + 3 + ... \end{align}

Furthermore, the 7^th partial sum of this sequence is:

(2)

\begin{align} \quad \sum_{n=1}^{7} a_n = 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28 \end{align}

Indices of a Series

Consider the sequence $\sum_{n=1}^{\infty} a_n$. At any point we can algebraically "alter" the indices to our desire where it may come in use. For example, we should note that the series listed above and the series $\sum_{m=5}^{\infty} a_{m-4}$ are in fact the same.

Changing indices sometimes may be useful in determining whether a series is convergent or divergent which we will subsequently discuss. For example, consider the series $\sum_{k=1}^{\infty} \frac{1}{k+2}$.

Now suppose that we let $n = k + 2$. Then it follows that $\sum_{k=1}^{\infty} \frac{1}{k+2} = \sum_{n=3}^{\infty} \frac{1}{n} = \sum_{n=1}^{\infty} \frac{1}{n} - 1 - \frac{1}{2}$. We note that in certain cases, either of these equivalent series may be more "useful" to us.

Convergence and Divergence of a Series

Definition: A series $\sum_{n=1}^{\infty} a_n$ is said to be Convergent to the Sum $s$ if sequence of partial sums $\{ s_n \}$ converges to $s$, that is $\lim_{n \to \infty} s_n = s$ and so $\sum_{n=1}^{\infty} a_n = s$. A series is said to be Divergent if the sequence of partial sums $\{ s_n \}$ is divergent.

We will subsequently look at various theorems regarding convergent and divergent series.