Taylor and Maclaurin Series

Series and Sequences

By the end of this lesson, you will be able to determine convergence of sequences, apply convergence tests to series, and work with power series and Taylor/Maclaurin series.

Sequences

Sequence: an ordered list of numbers: ${a_1, a_2, a_3, \ldots}$ or ${a_n}$.

Convergence: a sequence ${a_n}$ converges if $\lim_{n \to \infty} a_n = L$ (a finite number). Otherwise, it diverges.

Examples:

• $a_n = \dfrac{1}{n}$: converges to 0
• $a_n = n$: diverges to $\infty$
• $a_n = (-1)^n$: diverges (oscillates)
• $a_n = \dfrac{n}{n+1}$: $\lim_{n\to\infty} = 1$ → converges to 1

Infinite Series

Infinite series: the sum of all terms of a sequence:

$$\sum_{n=1}^{\infty} a_n = a_1 + a_2 + a_3 + \cdots$$

Partial sums: $S_N = \displaystyle\sum_{n=1}^N a_n$

The series converges if $\lim_{N \to \infty} S_N = L$; the sum is $L$.

Geometric Series

$$\sum_{n=0}^{\infty} ar^n = a + ar + ar^2 + \cdots$$

Converges if $|r| < 1$: sum = $\dfrac{a}{1-r}$

Diverges if $|r| \geq 1$.

Example: $\displaystyle\sum_{n=0}^{\infty} \dfrac{1}{3^n} = \dfrac{1}{1 - 1/3} = \dfrac{3}{2}$

Convergence Tests

Divergence Test (nth-Term Test)

If $\lim_{n \to \infty} a_n \neq 0$ (or doesn't exist), the series diverges.

If $\lim_{n \to \infty} a_n = 0$, the test is inconclusive (the series may still diverge).

Example: $\displaystyle\sum_{n=1}^{\infty} \dfrac{n}{n+1}$: $\lim_{n\to\infty} \dfrac{n}{n+1} = 1 \neq 0$ → diverges.

Integral Test

If $f$ is positive, continuous, and decreasing on $[1, \infty)$, then $\displaystyle\sum_{n=1}^{\infty} f(n)$ and $\displaystyle\int_1^{\infty} f(x), dx$ both converge or both diverge.

$p$-Series: $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^p}$ converges if $p > 1$; diverges if $p \leq 1$.

Special case: harmonic series $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n}$ diverges ($p = 1$).

Comparison Test