数列敛散性牢九门

Geometric Series Test

\[|r| < 1 \iff \sum a r^n \text{ converges} \]

\(p\)-Series Test

\[p > 1 \iff \sum \frac 1 {n^p} \text{ converges} \]

\(n\)th Term Test

\[\lim\limits_{n \to \infty} a_n \ne 0 \implies \sum a_n \text{ diverges} \]

Integral Test

(Nonnegative series)

If \(f(x)\) is a continuous, positive, decreasing function and \(f(n) = a_n\), then

\[\int_1^\infty f(x) dx \text{ and } \sum a_n \text{ both converge or diverge} \]

Comparison Test

(Nonnegative series)

\[\sum b_n \text{ converges} \land a_n \le b_n \implies \sum a_n \text{ converges} \]

\[\sum b_n \text{ diverges} \land a_n \ge b_n \implies \sum a_n \text{ diverges} \]

Limit Comparison Test

(Nonnegative series)

• Case 1: \(\lim\limits_{n \to \infty} \frac {a_n} {b_n} = L\) where \(0 < L < \infty\)

\[\sum a_n \text{ and } \sum b_n \text{ both converge or diverge} \]

• Case 2: \(\lim\limits_{n \to \infty} \frac {a_n} {b_n} = 0\)

\[\sum b_n \text{ converges} \implies \sum a_n \text{ converges} \]

• Case 3: \(\lim\limits_{n \to \infty} \frac {a_n} {b_n} = \infty\)

\[\sum b_n \text{ diverges} \implies \sum a_n \text{ diverges} \]

Ratio Test

(Nonnegative series)

Let \(\lim\limits_{n \to \infty} \frac {a_{n+1}} {a_n} = L\) if exists.

\[L < 1 \implies \sum a_n \text{ converges} \]

\[L > 1 \implies \sum a_n \text{ diverges} \]

Root Test

(Nonnegative series) 应该是吧，书上没说

Let \(\lim\limits_{n \to \infty} \sqrt[n]{a_n} = L\) if exists.

\[L < 1 \implies \sum a_n \text{ converges} \]

\[L > 1 \implies \sum a_n \text{ diverges} \]

Alternating Series Test

(\(a_n \ge 0\))

\[\begin{cases} a_{n+1} < a_n \\ \lim\limits_{n \to \infty} a_n = 0 \\ \end{cases} \implies \sum (-1)^n a_n \text{ converges}\]