1. What is an Infinite Series?

An infinite series is the sum of all the terms in a sequence:

$$a_1 + a_2 + a_3 + \dots = \sum_{n=1}^{\infty} a_n$$

We make sense of this by looking at partial sums:

$$s_n = a_1 + a_2 + \dots + a_n$$

• If $s_n \to S$ (a finite number) as $n \to \infty$, then the series converges and the sum is $S$.
• If not, the series diverges.

Takeaway: If the running total “settles down” to a number, the series converges.

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2. The Basic "nth-Term" Test (for Divergence)

A quick test: if the terms don’t shrink to 0, then the series can’t converge.

$$\boxed{\text{If } \lim_{n \to \infty} a_n \neq 0, \text{ the series diverges.}}$$

Takeaway: If the terms don’t go to 0, the series must diverge — no exceptions.

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3. Geometric Series

A geometric series looks like this:

$$a + ar + ar^2 + ar^3 + \dots = \sum_{n=1}^{\infty} ar^{n-1}$$

It converges only if $|r| < 1$, and then:

$$\sum_{n=1}^{\infty} ar^{n-1} = \frac{a}{1 - r}$$

Otherwise, it diverges.

Examples:

• $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$ → $a = \frac{1}{2}, r = \frac{1}{2} \Rightarrow \frac{1/2}{1 - 1/2} = 1$

• $0.\overline{23} = 0.23 + 0.0023 + 0.000023 + \dots$ → $a = 0.23, r = \frac{1}{100} \Rightarrow \frac{0.23}{1 - \frac{1}{100}} = \frac{23}{99}$

Takeaway: If you see a constant ratio between terms, you can find the exact sum.

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4. Algebra Rules for Series

If $\sum a_n$ and $\sum b_n$ both converge, then:

• $\sum (c \cdot a_n)$ converges
• $\sum (a_n + b_n)$ converges
• $\sum (a_n - b_n)$ converges

Takeaway: You can scale or combine convergent series and still get convergence.

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5. The Integral Test and p-Series

If $a_n = f(n)$, and $f(x)$ is positive, continuous, and decreasing, then:

$$\sum_{n=1}^{\infty} f(n) \text{ converges} \iff \int_{1}^{\infty} f(x) \, dx \text{ converges}$$

Example: p-Series

$$\sum_{n=1}^{\infty} \frac{1}{n^p}$$

• Converges if $p > 1$
• Diverges if $p \leq 1$

Takeaway: Integration can tell us whether a series adds up or not, especially when it's hard to use algebra.

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6. Comparison Tests

Use these when your series looks complicated but resembles a simpler one.

A. Direct Comparison Test

If:

• $0 \leq a_n \leq b_n$
• and $\sum b_n$ converges → then $\sum a_n$ also converges

(If $\sum b_n$ diverges and $a_n \geq b_n$, then $\sum a_n$ diverges too.)

B. Limit Comparison Test

If:

$$\lim_{n \to \infty} \frac{a_n}{b_n} = c \text{ where } 0 < c < \infty$$

Then $\sum a_n$ and $\sum b_n$ either both converge or both diverge.

Takeaway: If your series behaves “like” a known one at infinity, it’ll share the same fate.

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7. Summary: Strategy for Any Series

When you see a new infinite series:

1. nth-Term Test Does $a_n \to 0$?

   • If not → Diverges

2. Geometric? p-Series?

   • Recognize these immediately

3. Use the Integral Test

   • For positive, decreasing functions

4. Comparison Tests

   • Compare with simpler series

5. More advanced tools (next sections):

   • Ratio Test, Alternating Series Test, etc.

Final Thought: All tests are ways of asking: "Do the running totals approach a real number?"

Whether it’s comparing, integrating, or recognizing patterns, the goal is the same: convergence or divergence.