급수 수렴 판정법

1. Partial Sum

$${\{S_n=\sum _{i=1}^n{a}_i\}}_{n=1}^{\mathrm{\infty }}\mathrm{converges}\to \sum _{i=1}^n{a}_i\mathrm{converges}$$

$${\{S_n=\sum _{i=1}^n{a}_i\}}_{n=1}^{\mathrm{\infty }}\mathrm{diverges}\to \sum _{i=1}^n{a}_i\mathrm{diverges}$$

2. Cauchy Criterion

$$\mathrm{\forall }ϵ>0,\mathrm{\exists }Ns.t.$$

$$n>m>N\to \left|\sum _{i=m+1}^n{a}_i\right|<ϵ$$

3. Absolute Convergence

$$\sum _{n=1}^{\mathrm{\infty }}|a_n|\to c\Rightarrow \sum _{n=1}^{\mathrm{\infty }}{a}_n\to d$$

4. ($a_n>0$) Comparison Test

$a_n,b_n>0$ and $a_n\le b_n$ for all $n\ge N_0$,

$$\sum _{n=1}^{\mathrm{\infty }}{b}_n\mathrm{converges}\to \sum _{n=1}^{\mathrm{\infty }}{a}_n\mathrm{converges}$$

$$\sum _{n=1}^{\mathrm{\infty }}{a}_n\mathrm{diverges}\to \sum _{n=1}^{\mathrm{\infty }}{b}_n\mathrm{diverges}$$

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

$a_n,b_n>0$ and $\underset{n\to \mathrm{\infty }}{lim}{a}_n/b_n=c$

$\sum _{n=1}^{\mathrm{\infty }}{a}_n$ and $\sum _{n=1}^{\mathrm{\infty }}{b}_n$ are both convergent, or both divergent.

6. ($a_n>0$) Ratio Test

Suppose $a_n>0$.

1. $\underset{n\to \mathrm{\infty }}{lim sup}\frac{a_{n+1}}{a_n}<1$  $\to$  $\sum _{n=1}^{\mathrm{\infty }}{a}_n$ converges.

2. $\underset{n\to \mathrm{\infty }}{lim inf}\frac{a_{n+1}}{a_n}>1$  $\to$  $\sum _{n=1}^{\mathrm{\infty }}{a}_n$ diverges.

3. $\underset{n\to \mathrm{\infty }}{lim inf}\frac{a_{n+1}}{a_n}\le 1\le \underset{n\to \mathrm{\infty }}{lim sup}\frac{a_{n+1}}{a_n}$  $\to$  no conclusion.

7. ($a_n>0$) Root Test

Suppose $a_n>0$ and $\underset{n\to \mathrm{\infty }}{lim sup}{a}_n^{\frac{1}{n}}=\rho$

1. $\rho <1$  $\to$  $\sum _{n=1}^{\mathrm{\infty }}{a}_n$ converges.

2. $\rho >1$  $\to$  $\sum _{n=1}^{\mathrm{\infty }}{a}_n$ diverges.

3. $\rho =1$  $\to$  no conclusion.

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

$\sum _{n=1}^{\mathrm{\infty }}{a}_n$ converges iff $\sum _{n=1}^{\mathrm{\infty }}{2}^n{a}_{2^n}$ converges.

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

If $a_n\to 0$ as $n\to \mathrm{\infty }$, then $\sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n-1}{a}_n$ converges.