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 $A ≤ B$. $\blacksquare$