Basic Properties of Convergent Infinite Series

# Basic Properties of Convergent Infinite Series

Recall from the Convergence and Divergence of Infinite Series page that a series $\displaystyle{\sum_{n=1}^{\infty} a_n}$ is said to be convergent if the corresponding sequence of partial sums $(s_n)_{n=1}^{\infty}$ converges (where of course, $\displaystyle{s_n = \sum_{k=1}^{n} a_k}$).

Furthermore, we said that $\displaystyle{\sum_{n=1}^{\infty} a_n}$ diverges if it does not converge. We will now look at some very basic properties regarding convergent infinite series.

 Theorem 1: If $\displaystyle{\sum_{n=1}^{\infty} a_n}$ and $\displaystyle{\sum_{n=1}^{\infty} b_n}$ converge to $A$ and $B$ respectively, then $\displaystyle{\sum_{n=1}^{\infty} [a_n + b_n]}$ converges to $A + B$.
• Proof: Let $(s_n)_{n=1}^{\infty}$ and $(s_n')_{n=1}^{\infty}$ be the corresponding sequence of partial sums for $\displaystyle{\sum_{n=1}^{\infty} a_n}$ and $\displaystyle{\sum_{n=1}^{\infty} b_n}$ respectively. Since these two series converge to $A$ and $B$ respectively, we have that:
(1)
\begin{align} \quad \lim_{n \to \infty} s_n = A \quad \mathrm{and} \quad \lim_{n \to \infty} s_n' = B \end{align}
• Let $s_n^* = s_n + s_n'$. Then $\displaystyle{s_n* = \sum_{k=1}^{n} a_k + \sum_{k=1}^{n} b_k = \sum_{k=1}^{n} [a_k + b_k]}$ is the general term for the sequence of partial sums for the series $\displaystyle{\sum_{n=1}^{\infty} [a_n + b_n]}$ and moreover:
(2)
\begin{align} \quad \lim_{n \to \infty} s_n^* = \lim_{n \to \infty} [s_n + s_n'] = \lim_{n \to \infty} s_n + \lim_{n \to \infty} s_n' = A + B \end{align}
• Therefore $\displaystyle{\sum_{n=1}^{\infty} [a_n + b_n]}$ converges to $A + B$. $\blacksquare$

It is EXTREMELY important to note that the converse to Theorem 1 is not true in general. If $\displaystyle{\sum_{n=1}^{\infty} [a_n + b_n]}$ converges then it may be that $\displaystyle{\sum_{n=1}^{\infty} a_n}$ and/or $\displaystyle{\sum_{n=1}^{\infty} b_n}$ may diverge.

For example, let $a_n = 1$ and $b_n = -1$ for all $n \in \mathbb{N}$. Then $\displaystyle{\sum_{n=1} [a_n + b_n] = \sum_{n=1}^{\infty} 0 = 0}$, however it should be rather clear that $\displaystyle{\sum_{n=1}^{\infty} a_n = \sum_{n=1}^{\infty} 1 = \infty}$ and $\displaystyle{\sum_{n=1}^{\infty} b_n = \sum_{n=1}^{\infty} -1 = -\infty}$ do not converge.

 Theorem 2: The series $\displaystyle{\sum_{n=1}^{\infty} a_n}$ converges to $A$ if and only if for all $c \in \mathbb{R}$, $c \neq 0$ we have that the series $\displaystyle{\sum_{n=1}^{\infty} ca_n}$ converges to $cA$.
• Proof: $\Rightarrow$ Suppose that $\displaystyle{\sum_{n=1}^{\infty} a_n}$ converges to $A$. If $(s_n)_{n=1}^{\infty}$ is the corresponding sequence of partial sums for this series then $(s_n)_{n=1}^{\infty}$ converges to $A$. Let $s_n^* = cs_n$ for all $n \in \mathbb{N}$. Then $(s_n^*)_{n=1}^{\infty}$ is the sequence of partial sums for the series $\displaystyle{\sum_{n=1}^{\infty} ca_n}$.
• Furthermore, $\displaystyle{\lim_{n \to \infty} s_n^* = \lim_{n \to \infty} cs_n = c \lim_{n \to \infty} s_n = cA}$. Therefore $\displaystyle{\sum_{n=1}^{\infty} ca_n}$ converges to $cA$. $\blacksquare$
• $\Leftarrow$ Let $c \in \mathbb{R}$ and suppose that $\sum_{n=1}^{\infty} ca_n$ converges to $cA$. If $(s_n)_{n=1}^{\infty}$ is the corresponding sequence of partial sums for this series then $(s_n)_{n=1}^{\infty}$ converges to $cA$. Let $\displaystyle{s_n^* = \frac{1}{c} s_n}$ for all $n \in \mathbb{N}$. Then $(s_n^*)_{n=1}^{\infty}$ is the sequence of partial sums for the series $\displaystyle{\sum_{n=1}^{\infty} a_n}$.
• Furthermore $\displaystyle{\lim_{n \to \infty} s_n^* = \lim_{n \to \infty} \frac{1}{c} s_n = \frac{1}{c} \lim_{n \to \infty} s_n = \frac{1}{c} \cdot cA = A}$. Therefore $\displaystyle{\sum_{n=1}^{\infty} a_n}$ converges to $A$. $\blacksquare$