Series Convergence and Divergence Proofs

# Series Convergence and Divergence Proofs

Recall that a series $\sum_{n=1}^{\infty} a_n$ is said to Converge to the sum $s$ if the sequence of partial sums $\{ s_n \}$ where $s_n = a_1 + a_2 + ... + a_n = sum_{i=1}^{n} a_i$ converges to the sum $s$, and the series is said to Diverge if it does not converge. We will now look at some examples of proving/disproving statements regarding the convergence/divergence of a series.

## Example 1

Prove or disprove: If the series $\sum_{n=1}^{\infty} a_n$ and $\sum_{n=1}^{\infty} b_n$ both diverge then $\sum_{n=1}^{\infty} (a_n + b_n)$ diverges.

If $\sum_{n=1}^{\infty} a_n$ and $\sum_{n=1}^{\infty} b_n$ both diverge, then their sequences of partial sums, say $\{ s_n \}$ and $\{ s'_n \}$ both diverge. Thus, we need to determine whether the sequence of partial sums $\{ s_n + s'_n \}$ must also diverge.

Consider the series whose partial sums form the sequences $\{ s_n \} = \{ (-1)^n \} = \{ -1, 1, -1, 1, ... \}$ and $\{ s_n' \} = \{ (-1)^{n+1} \} = \{ 1, -1, 1, -1, ... \}$, both of which diverge. However, notice that the sequence $\{ s_n + s'_n \} = \{ 0, 0, ... \}$ converges

Therefore the statement is false in general.

## Example 2

Prove or disprove: If $a_n = 0$ for every $n \in \mathbb{N}$ then $\sum_{n=1}^{\infty} a_n$ converges.

This statement is true. If $a_n = 0$ for every $n \in \mathbb{N}$ then the sequence of terms for this series is $\{ a_n \} = \{ 0, 0, ... \}$, and so the sequence of partial sums is $\{ s_n \} = \{ 0 \}$ and $\lim_{n \to \infty} s_n = \lim_{n \to \infty} 0 = 0$.

## Example 3

Prove or disprove: If $a_n ≥ c > 0$ for every $n \in \mathbb{N}$ then $\sum_{n=1}^{\infty} a_n$ diverges to infinity.

Suppose that $a_n ≥ c > 0$ for every $n \in \mathbb{N}$, and consider the sequence of partial sums $\{ s_n \}$. Notice that $s_1 = a_1 ≥ c > 0$, $s_2 = a_1 + a_2 ≥ 2c > 0$, …, $s_n = a_1 + a_2 + ... + a_n ≥ nc > 0$.

Now we have that $\lim_{n \to \infty} nc = \infty$, and since $nc ≤ s_n$ for all $n \in \mathbb{N}$ then $\lim_{n \to \infty} s_n = \infty$ by a comparison theorem for sequences.

## Example 4

Prove or disprove: If $\sum_{n=1}^{\infty} a_n$ converges then $\sum_{n=1}^{\infty} \frac{1}{a_n}$ must diverge to infinity.

Consider the series $\sum_{n=1}^{\infty} \left ( -\frac{1}{2} \right )^n$ (note that here, $a_n = \left ( - \frac{1}{2} \right )^n$. This series converges as a geometric series whose common ratio $r = \frac{1}{2}$ is such that $\mid r \mid < 1$. Now note that:

(1)
\begin{align} \quad \sum_{n=1}^{\infty} \frac{1}{a_n} = \sum_{n=1}^{\infty} (-2)^n \end{align}

This series does indeed diverge, however, it does not diverge to infinity.