급수 수렴 판정법

1. Partial Sum

$$ \left\{S_n=\sum_{i=1}^n a_i\right\}_{n=1}^\infty\quad \mathrm{converges}\quad\rightarrow \quad \sum_{i=1}^n a_i\quad \mathrm{converges} $$

$$\left\{S_n=\sum_{i=1}^n a_i\right\}_{n=1}^\infty\quad \mathrm{diverges}\quad\rightarrow \quad \sum_{i=1}^n a_i \quad \mathrm{diverges}$$

2. Cauchy Criterion

$$\forall \epsilon >0,\;\exists N \quad s.t.$$

$$n>m>N\;\rightarrow\;\left|\sum_{i=m+1}^na_i\right|<\epsilon$$

3. Absolute Convergence

$$\sum_{n=1}^\infty |a_n| \;\rightarrow\;c\quad \Rightarrow \quad \sum_{n=1}^\infty a_n \;\rightarrow\;d$$

4. (\(a_n>0\)) Comparison Test

\(a_n, b_n > 0\) and \(a_n \leq b_n\) for all \(n\geq N_0\),

$$\sum_{n=1}^\infty b_n \;\mathrm{converges}\quad\rightarrow\quad \sum_{n=1}^\infty a_n\;\mathrm{converges}$$

$$\sum_{n=1}^\infty a_n \;\mathrm{diverges}\quad\rightarrow\quad \sum_{n=1}^\infty b_n\;\mathrm{diverges}$$

5. (\(a_n>0\)) Fractional Comparison Test

\(a_n, b_n > 0\) and \(\lim_{n\rightarrow\infty}a_n/b_n=c\)

\(\sum_{n=1}^\infty a_n\) and \(\sum_{n=1}^\infty b_n\) are both convergent, or both divergent.

6. (\(a_n>0\)) Ratio Test

Suppose \(a_n>0\).

1. \(\limsup_{n\rightarrow \infty}\frac{a_{n+1}}{a_n}<1\)  \(\rightarrow\)  \(\sum_{n=1}^\infty a_n\) converges.

2. \(\liminf_{n\rightarrow \infty}\frac{a_{n+1}}{a_n}>1\)  \(\rightarrow\)  \(\sum_{n=1}^\infty a_n\) diverges.

3. \(\liminf_{n\rightarrow \infty}\frac{a_{n+1}}{a_n} \leq 1 \leq \limsup_{n\rightarrow \infty}\frac{a_{n+1}}{a_n}\)  \(\rightarrow\)  no conclusion.

7. (\(a_n>0\)) Root Test

Suppose \(a_n>0\) and \(\limsup_{n\rightarrow\infty}a_n^{\frac{1}{n}}=\rho\)

1. \(\rho<1\)  \(\rightarrow\)  \(\sum_{n=1}^\infty a_n\) converges.

2. \(\rho>1\)  \(\rightarrow\)  \(\sum_{n=1}^\infty a_n\) diverges.

3. \(\rho=1\)  \(\rightarrow\)  no conclusion.

8. (\(a_n>0\), monotone decreasing) Condensation Test

\(\sum_{n=1}^\infty a_n\) converges iff \(\sum_{n=1}^\infty 2^na_{2^n}\) converges.

9. (\(a_n>0\), monotone decreasing) Alternating Series Test

If \(a_n\rightarrow 0\) as \(n\rightarrow\infty\), then \(\sum_{n=1}^\infty (-1)^{n-1}a_n\) converges.