Series and Sequences

1. What Are Sequences and Series?

• Sequence: A list of numbers in order
  $a_1, a_2, a_3, \dots$

• Series: The sum of a sequence
  $\sum_{n=1}^{\infty} a_n$

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2. Divergence Test (Test for Divergence)

If
$\lim_{n \to \infty} a_n \neq 0$
or does not exist → the series diverges

Example

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

Solution: $\lim_{n \to \infty} \frac{n}{n+1} = 1 \neq 0$
❌ Diverges

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3. Integral Test

If:

• $f(n) = a_n$ is positive, continuous, decreasing

Then: $\sum a_n \text{ and } \int_1^\infty f(x)\,dx$
either both converge or both diverge

Example

$\sum_{n=1}^{\infty} \frac{1}{n^2}$

Solution: $\int_1^\infty \frac{1}{x^2} dx = \left[-\frac{1}{x}\right]_1^\infty = 1$
✅ Converges → Series converges

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4. p-Series Test

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

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

Example

$\sum_{n=1}^{\infty} \frac{1}{n^{1.5}}$

$p = 1.5 > 1$ → ✅ Converges

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5. Direct Comparison Test

Compare with a known series:

• If $0 \le a_n \le b_n$ and $\sum b_n$ converges → $\sum a_n$ converges
• If $a_n \ge b_n$ and $\sum b_n$ diverges → $\sum a_n$ diverges

Example

$\sum_{n=1}^{\infty} \frac{1}{n^2 + 1}$

Compare: $\frac{1}{n^2+1} \le \frac{1}{n^2}$

Since $\sum \frac{1}{n^2}$ converges → ✅ Converges

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6. Limit Comparison Test

If: $\lim_{n \to \infty} \frac{a_n}{b_n} = c$
where $0 < c < \infty$, both series behave the same

Example

$\sum_{n=1}^{\infty} \frac{3n+1}{n^2+2}$

Compare with: $\frac{1}{n}$

Compute: $\lim_{n\to\infty} \frac{\frac{3n+1}{n^2+2}}{\frac{1}{n}} = \lim_{n\to\infty} \frac{n(3n+1)}{n^2+2} = 3$

Since $\sum \frac{1}{n}$ diverges → ❌ Diverges

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7. Ratio Test

Compute: $L = \lim_{n\to\infty} \left| \frac{a_{n+1}}{a_n} \right|$

• $L < 1$ → Converges
• $L > 1$ → Diverges
• $L = 1$ → Inconclusive

Example

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

$\frac{a_{n+1}}{a_n} = \frac{1/(n+1)!}{1/n!} = \frac{1}{n+1}$

$\lim_{n\to\infty} \frac{1}{n+1} = 0 < 1$
✅ Converges

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8. Geometric Series

Form: $\sum_{n=0}^{\infty} ar^n$

• Converges if $|r| < 1$
• Sum: $\frac{a}{1-r}$

Example

$\sum_{n=0}^{\infty} 2\left(\frac{1}{3}\right)^n$

$a=2$, $r=\frac{1}{3}$

$S = \frac{2}{1 - \frac{1}{3}} = \frac{2}{\frac{2}{3}} = 3$
✅ Converges, sum = 3

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9. Alternating Series Test (Leibniz Test)

For: $\sum_{n=1}^{\infty} (-1)^n a_n$

Converges if:

1. $a_n$ decreases
2. $\lim_{n \to \infty} a_n = 0$

Example

$\sum_{n=1}^{\infty} \frac{(-1)^n}{n}$

• Decreasing ✔
• Limit = 0 ✔

✅ Converges

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Quick Summary

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