Convergence/Divergence of Series for MATH 141

When Does an Infinite Sum Make Sense?

Adding infinitely many numbers sounds impossible — and sometimes it is! The series $1 + 2 + 3 + 4 + \cdots$ grows forever. But $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots$ adds up to exactly 1. The difference? Convergence.

A series converges if its partial sums approach a finite number. It diverges if they don't.

The Definition

Given a series $\displaystyle\sum_{n=1}^{\infty} a_n$, let $S_N$ be the $N$th partial sum:

$$S_N = a_1 + a_2 + \cdots + a_N$$

• If $\displaystyle\lim_{N \to \infty} S_N = L$ (a finite number), the series converges to $L$
• If the limit doesn't exist or is infinite, the series diverges

The Divergence Test (nth Term Test)

This is the simplest and most important first check for any series.

Theorem: If $\displaystyle\sum_{n=1}^{\infty} a_n$ converges, then $\displaystyle\lim_{n \to \infty} a_n = 0$.

Contrapositive (the useful form):

$$\text{If } \lim_{n \to \infty} a_n \neq 0, \text{ then } \sum a_n \text{ diverges.}$$

⚠️ Critical Warning

The Divergence Test can only prove divergence, never convergence!

If $\lim_{n \to \infty} a_n = 0$, you learn nothing — the series might converge or diverge.

Example: $\displaystyle\sum \frac{1}{n}$ has $\lim_{n \to \infty} \frac{1}{n} = 0$, but the series diverges (it's the harmonic series).

Finding the Sum Using Partial Sums

When you can find a formula for $S_N$, you can determine convergence by taking the limit.

Finding $a_n$ from $S_n$

Sometimes you're given the partial sum formula and need to find the terms.

Key relationship:

$$a_n = S_n - S_{n-1} \quad \text{for } n \geq 2$$

$$a_1 = S_1$$

Properties of Convergent Series

If $\displaystyle\sum a_n = A$ and $\displaystyle\sum b_n = B$ (both converge), then:

Warning: These only apply when both series converge!

Adding or Removing Finite Terms

Adding or removing a finite number of terms doesn't change whether a series converges or diverges (though it changes the sum).

$$\sum_{n=1}^{\infty} a_n = a_1 + a_2 + \cdots + a_k + \sum_{n=k+1}^{\infty} a_n$$

If one side converges, so does the other.

Quick Summary: Convergence vs Divergence

Common Mistakes and Misunderstandings

❌ Mistake: Using the Divergence Test to prove convergence

Wrong: "$\lim_{n \to \infty} \frac{1}{n} = 0$, so by the Divergence Test, $\sum \frac{1}{n}$ converges."

Why it's wrong: The Divergence Test can ONLY prove divergence. When the limit is 0, the test tells you absolutely nothing.

Correct: When $\lim a_n = 0$, say "The Divergence Test is inconclusive" and use another test.

❌ Mistake: Confusing $a_n \to 0$ with $\sum a_n$ converging

Wrong: "The terms get smaller, so the series must converge."

Why it's wrong: The harmonic series $\sum \frac{1}{n}$ has terms going to 0, but it diverges! Terms approaching 0 is necessary but not sufficient for convergence.

Correct: $a_n \to 0$ is required for convergence, but you need additional tests to confirm it.

❌ Mistake: Forgetting $a_1 = S_1$ when finding terms

Wrong: Using $a_n = S_n - S_{n-1}$ for $n = 1$.

Why it's wrong: $S_0$ is not defined (there's no "sum of zero terms" in most contexts). The formula only works for $n \geq 2$.

Correct: Always find $a_1$ directly as $a_1 = S_1$, then use $a_n = S_n - S_{n-1}$ for $n \geq 2$.

❌ Mistake: Thinking convergent series can be rearranged freely

Wrong: "Since $\sum a_n = L$, I can add the terms in any order and still get $L$."

Why it's wrong: This is only true for absolutely convergent series. For conditionally convergent series (like the alternating harmonic series), rearranging terms can give ANY sum you want — or even diverge!

Correct: Be careful with rearrangements. For now, keep terms in their original order.