Summary of Test of Series

1. Absolutely Converge하면 Converge??

• by Squeeze Theorem for Sequences,
  if $lim_{n \rightarrow \infty} |a_n| = 0,$ then $lim_{n \rightarrow \infty} a_n = 0$

2. 단순히 수열은 l'Hospital's Rule을 적용할 수 없지만, 그에 상응하는 $f(x)$에는 적용할 수 있다.

3. geometric series는 common ratio $r$에 의해서 수렴 여부가 결정된다.

Test for Divergence

4. if $\sum_{n=1}^\infty a_n$ is convergent, then $lim_{n \rightarrow \infty} \ a_n = 0$
   but the converse of this is not always true in general

Integral Test

5. If $\int_{1}^\infty f(x)\,\mathrm{d}x$ is convergent, then $\sum_{n=1}^\infty a_n$ is convergent.

• $f$ don't need to be always decreasing.
  What is important is that $f$ be ultimately decreasing.
• $\int_{1}^\infty f(x)\,\mathrm{d}x \ne \sum_{n=1}^\infty a_n$

6. The p-series $\sum_{n=1}^\infty {1 \over n^p}$ is convergent if $p>1$.

Remainder Estimate for Integral Test

7. $\int_{n+1}^\infty f(x)\,\mathrm{d}x$ $\leq R_n \leq$ $\int_{n}^\infty f(x)\,\mathrm{d}x$

• $f$ is a continuous, positive, decreasing function for $x \ge n$

Direct Comparison Test

8. $\sum b_n$이 수렴하고, $a_n \leq b_n$ for all n, $\sum a_n$이 수렴한다.
   $\sum b_n$이 발산하고, $a_n \geq b_n$ for all n, $\sum a_n$이 발산한다.

Limit Comparison Test

9. $\lim_{n \to \infty} {a_n / b_n} = c$이고 $b_n$이 수렴하면, $a_n$도 수렴.

Alternating Series Test

10. 만약 Alternating Seires의 $b_n$이
    $(i)$ $b_{n+1} \leq b_n$ for all $n$
    $(ii)$ $\lim_{n \to \infty} b_n = 0$
    를 만족시키면, 수렴(converge)한다.

• $Proof$ : $s_{2n}$ is monotonic(increasing) and bounded, so convergent.
• What really matters is ${b_n}$is eventually decreasing.

Alternating Series Estimation Theorem

11. Alternating Series Test를 통과한다면,
    $\left\vert R_n \right\vert = \left\vert s - s_n\right\vert \leq b_{n+1}$

12. Absolutely convergent / Conditionally convergent :
    if series is absolutely convergent, it also converges.
    if series is convergent, but not absolutely convergent.

Rearrangement

13. Series가 absolutely convergent하면 가능하지만
    Series가 conditional convergent하면 불가능하다.

Ratio Test

14. $(i)$ if $lim_{n\rightarrow\infty} |{a_{n+1}\over a_{n}}| = L < 1$, the series is absolutely convergent. (therfore convergent).
    $(ii)$ If $lim_{n\rightarrow\infty} |{a_{n+1}\over a_{n}}| = 1$, The Ratio Test is inconclusive. need to check.

Root Test

15. If $a_n$ is the form of $\{b_n\}^n$, Test the value of $lim_{n\rightarrow\infty} \sqrt[n]{|a_n|} = L$