1.

\[\sum_{n=1}^{\infty} a_n = \lim_{n \to \infty} S_n\]

• \(a_n\) is the \(n^{\text{th}}\) term
• \(S_n\) is the sum of the first \(n\) terms (partial sum)

If

\[\lim_{n \to \infty} S_n \text{ converges, then } \sum_{n=1}^{\infty} a_n \text{ converges}\]

If

\[\lim_{n \to \infty} S_n \text{ diverges, then } \sum_{n=1}^{\infty} a_n \text{ diverges}\]

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2.

Geometric Infinite Series

• If \(|r| \geq 1\), the series diverges
• If \(|r| < 1\), the series converges

\[\sum_{n=1}^{\infty} ar^n = \frac{ar^k}{1-r}\]

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3.

\(n^{\text{th}}\) Term Test for Divergence

If

\[\lim_{n \to \infty} a_n \neq 0, \quad \text{then} \sum_{n=1}^{\infty} a_n \text{ diverges}\]

If

\[\lim_{n \to \infty} a_n = 0, \quad \text{then} \sum_{n=1}^{\infty} a_n \text{ may converge or diverge}\]

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4.

Integral Test for Convergence

If \( f \) is a positive, continuous, and decreasing function for \( x \geq k \), and \( a_n = f(n) \), then
\[\sum_{n=k}^{\infty} a_n \quad \text{and} \quad \int_{k}^{\infty} f(x) \, dx\]
both converge or both diverge.

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5.

p-Series

Let \( p \) be a positive constant of the series:
\[\sum_{n=1}^{\infty} \frac{1}{n^p} = \frac{1}{1^p} + \frac{1}{2^p} + \frac{1}{3^p} + \cdots\]

• The series converges if \( p > 1 \)
• The series diverges if \( 0 < p \leq 1 \)

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6.

Comparison Test

Let \( 0 < a_n \leq b_n \) for all \( n \).

• If
  \[\sum_{n=1}^{\infty} b_n \text{ converges, then } \sum_{n=1}^{\infty} a_n \text{ also converges}\]
• If
  \[\sum_{n=1}^{\infty} a_n \text{ diverges, then } \sum_{n=1}^{\infty} b_n \text{ also diverges}\]

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7.

Limit Comparison Test

If \( a_n > 0 \), \( b_n > 0 \), and \[ \lim_{n \to \infty} \frac{a_n}{b_n} = L \quad \text{(where \( L \) is finite and positive)} \] then \[ \sum_{n=1}^{\infty} a_n \quad \text{and} \quad \sum_{n=1}^{\infty} b_n \] either both converge or both diverge.

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8.

Alternating Series Test

If \( a_n > 0 \), then the alternating series \[ \sum_{n=1}^{\infty} (-1)^n a_n \quad \text{and} \quad \sum_{n=1}^{\infty} (-1)^{n+1} a_n \] converge if both of the following are true:

1. \[ \lim_{n \to \infty} a_n = 0 \] 2. \[ |a_{n+1}| \leq |a_n| \]

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9.

Ratio Test for Convergence

For a series \( \sum_{n=1}^{\infty} a_n \) with positive terms, consider: \[ \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \begin{cases} < 1 & \text{Converges} \\ > 1 & \text{Diverges} \\ = 1 & \text{Use another test} \end{cases} \]

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10.

Absolute and Conditional Convergence

1) Converges Absolutely:

If \[ \sum_{n=1}^{\infty} |a_n| \text{ converges,} \] then the original series \[ \sum_{n=1}^{\infty} a_n \text{ converges Absolutely.} \]

2) Converges Conditionally:

If \[ \sum_{n=1}^{\infty} |a_n| \text{ diverges,} \] but the original series \[ \sum_{n=1}^{\infty} a_n \text{ converges,} \] then the series converges conditionally.

3) Divergent:

Both \[ \sum_{n=1}^{\infty} |a_n| \quad \text{and} \quad \sum_{n=1}^{\infty} a_n \] diverge. then the series is Divergent