Sequences and Series

Given by: a_n = a₁ r^n-1, where a₁ is the first term and r is the common ratio.
Or a_n = a₀rⁿ
Or a_n = a₂ r^n-2

10.2

A geometric series converges if |r| < 1.

$$\sum_{n=k}^{\infty}ar^n=\frac{ar^k}{1-r}$$
where ar^k is the first term of the series.

10.2

A geometric series diverges if |r| ≥ 1.

10.2

If $$\lim_{n\rightarrow\infty}a_n\ne0$$ then $$\sum_{n=1}^{\infty}a_n$$ diverges

10.3

If f is positive, continuous, and decreasing function for x ≥ k, and a_n = f(x), then

$$\sum_{n=k}^{\infty}a_n$$ and $$\int_k^{\infty}f\left(x\right)dx$$
both converge or both diverge.

10.4

A p-series is $$\sum_{n=1}^{\infty}\frac{1}{n^p}=\frac{1}{1^p}+\frac{1}{2^p}+\frac{1}{3^p}+...$$ converges if p > 1, diverges if p ≤ 1.

10.5

$$\sum_{n=1}^{\infty}\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+...$$

always DIVERGES

10.5

Compare to a known series:

If 0 ≤ aₙ ≤ bₙ and ∑bₙ converges, then ∑aₙ converges.

If aₙ ≥ bₙ ≥ 0 and ∑bₙ diverges, then ∑aₙ diverges.

10.6

If a_n > 0, b_n > 0 and $$\lim_{n\rightarrow\infty}\ \frac{a_n}{b_n}=\ L$$ where L is finite and positive, then

$$\sum_{n=1}^{\infty}a_n$$ and $$\sum_{n=1}^{\infty}b_n$$

either BOTH converge or diverge.

10.6

If terms alternate in sign, decrease in magnitude, and approach 0, the series converges.

10.7