Note 11 - 07/04/2023
TABLE OF CONTENTS
Theorem
- If the series $\sum_{n=1}^{\infty} a_n$ is convergent, then $\lim_{n \to \infty} a_n=0$.
Steps for convergent or divergent test
- If $\lim_{n \to \infty} a_n \neq 0$ then STOP. Summary: the series diverges.
If $\lim_{n \to \infty} S_n = S \in \mathbb{R}$ then STOP. Summary: the series converges at $S$.
If $\lim_{n \to \infty} S_n$ does not exist or approaches $\pm \infty$ then STOP. Summary: the series diverges.
- Use convergent tests:
Non-negative series: $\sum_{n=1}^{\infty} a_n$ with $a_n \geq 0$
Comparison test:
If $0 \leq a_n < b_n$ then
- $\sum_{n=1}^{\infty} a_n$ diverges $\Rightarrow$ $\sum_{n=1}^{\infty} b_n$ diverges.
- $\sum_{n=1}^{\infty} b_n$ converges $\Rightarrow$ $\sum_{n=1}^{\infty} a_n$ converges.
Limit test:
If $\lim_{n \to \infty} \dfrac{a_n}{b_n} = K \neq 0, \infty$ then $\sum_{n=1}^{\infty} a_n$ and $\sum_{n=1}^{\infty} b_n$ have identical property in convergent testing.
Integral test:
If $f(n)=a_n$ is a continuous and decreasing function on $[1,\infty)$ then $\int_1^\infty f(x) \ dx$ and $\sum_{n=1}^{\infty} a_n$ have identical property in convergent testing.
- Alternating series (aka Leibniz criterion): $\sum_{n=1}^\infty (-1)^nb_n=-b_1+b_2-b_3+b_4-…$ with $b_n>0$
- If $\{b_n\}$ decreases and $\displaystyle \lim_{n \to \infty} b_n=0$ then $\sum_{n=1}^\infty (-1)^nb_n$ converges.
- If $\{b_n\}$ increases and $\displaystyle \lim_{n \to \infty} b_n \neq 0$ then $\sum_{n=1}^\infty (-1)^nb_n$ diverges.
- Any series:
Absolute convergence:
If $\sum_{n=1}^\infty \vert a_n \vert$ converges then $\sum_{n=1}^\infty a_n$ converges absolutely.
Ratio test (aka d’Alembert criterion)
If $\displaystyle \lim_{n \to \infty} \left\vert \frac{a_{n+1}}{a_n} \right\vert = D$ then
- If $D>1$ then the series diverges.
- If $D<1$ then the series converges.
Root test (aka Cauchy criterion)
If $\displaystyle \lim_{n \to \infty} \sqrt[n]{\left\vert a_n \right\vert} = C$ then
- If $C>1$ then the series diverges.
- If $C<1$ then the series converges.