Properties of Convergent Series

Properties of Convergent Series

We will now look at some very important properties of convergent series, many of which follow directly from the standard limit laws for sequences.

 Theorem 1: Let $\sum_{n=1}^{\infty} a_n$ be convergent to the sum $A$ and let $\sum_{n=1}^{\infty} b_n$ be convergent to the sum $B$. Then the series $\sum_{a_n + b_n}$ is convergent to the sum $A + B$.
• Proof: Suppose that $\sum_{n=1}^{\infty} a_n = A$ and $\sum_{n=1}^{\infty} b_n = B$. Since both of these series are convergent, it follows that their sequences of partial sums are convergent, that is $\lim_{n\to \infty} s_n = A$ and that $\lim_{n \to \infty} s_n' = B$. By limit laws since both of these sequences are convergent it follows that $\lim_{n \to \infty} \left ( a_n + b_n \right ) = A + B$ which implies $\sum_{n=1}^{\infty} a_n + b_n = A + B$. $\blacksquare$
 Theorem 2: Let $\sum_{n=1}^{\infty} a_n$ be convergent to the sum $A$ and let $\sum_{n=1}^{\infty} b_n$ be convergent to the sum $B$. Then the series $\sum_{n=1}^{\infty} a_n - b_n$ is convergent to the difference $A - B$.
• Proof: Suppose that $\sum_{n=1}^{\infty} a_n = A$ and $\sum_{n=1}^{\infty} b_n = B$. Since both of these series are convergent, it follows that their sequences of partial sums are convergent, that is $\lim_{n\to \infty} s_n = A$ and that $\lim_{n \to \infty} s_n' = B$. By limit laws since both of these sequences are convergent it follows that $\lim_{n \to \infty} \left ( a_n - b_n \right ) = A - B$ which implies $\sum_{n=1}^{\infty} a_n - b_n = A - B$. $\blacksquare$
 Theorem 3: Let $\sum_{n=1}^{\infty} a_n$ be a convergent series to the sum $A$ and let $k$ be a constant. Then $\sum_{n=1}^{\infty} ka_n$ is also a convergent series to the sum $kA$.
• Proof: Suppose that $\sum_{n=1}^{\infty} a_n = A$ and let $\{ s_n \}$ be the sequence of partial sums for this series. Then $\lim_{n \to \infty} s_n = A$. Since this sequence is convergent, it follows by the limit laws that $\lim_{n \to \infty} ks_n = kA$ which implies that that $\sum_{n=1}^{\infty} ka_n = kA$. $\blacksquare$
 Theorem 4: Let $\sum_{n=1}^{\infty} a_n$ be a convergent convergent series to $A$ and let $\sum_{n=1}^{\infty} b_n$ be a convergent series to $B$. If $a_n ≤ b_n$ for all $n \in \mathbb{N}$, then $A ≤ B$.
• Proof: Suppose that $\sum_{n=1}^{\infty} a_n = A$ and $\sum_{n=1}^{\infty} b_n = B$ such that $a_n ≤ b_n$ for all $n \in \mathbb{N}$. We know that $\lim_{n \to \infty} s_n = A$ and $\lim_{n \to \infty} s_n' = B$. Since $a_n ≤ b_n$ for all $n \in \mathbb{N}$ we deduce that $s_n = a_1 + a_2 + ... + a_n ≤ b_1 + b_2 + ... + b_n = s_n'$ for all $n \in \mathbb{N}$, and so $A ≤ B$. $\blacksquare$