Ch. 5 Key Concepts - Calculus Volume 2

To determine the convergence of a sequence given by an explicit formula $a_{n} = f (n),$ we use the properties of limits for functions.
If ${a_{n}}$ and ${b_{n}}$ are convergent sequences that converge to $A$ and $B,$ respectively, and $c$ is any real number, then the sequence ${c a_{n}}$ converges to $c \cdot A,$ the sequences ${a_{n} \pm b_{n}}$ converge to $A \pm B,$ the sequence ${a_{n} \cdot b_{n}}$ converges to $A \cdot B,$ and the sequence ${a_{n} / b_{n}}$ converges to $A / B,$ provided $B \neq 0 .$
If a sequence is bounded and monotone, then it converges, but not all convergent sequences are monotone.
If a sequence is unbounded, it diverges, but not all divergent sequences are unbounded.
The geometric sequence ${r^{n}}$ converges if and only if $| r | < 1$ or $r = 1 .$

Given the infinite series

$\sum_{n = 1}^{\infty} a_{n} = a_{1} + a_{2} + a_{3} + ⋯$

and the corresponding sequence of partial sums ${S_{k}}$ where

$S_{k} = \sum_{n = 1}^{k} a_{n} = a_{1} + a_{2} + a_{3} + ⋯ + a_{k},$

the series converges if and only if the sequence ${S_{k}}$ converges.
The geometric series $\sum_{n = 1}^{\infty} a r^{n - 1}$ converges if $| r | < 1$ and diverges if $| r | \geq 1 .$ For $| r | < 1,$

$\sum_{n = 1}^{\infty} a r^{n - 1} = \frac{a}{1 - r} .$
The harmonic series

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

diverges.
A series of the form $\sum_{n = 1}^{\infty} [b_{n} - b_{n + 1}] = [b_{1} - b_{2}] + [b_{2} - b_{3}] + [b_{3} - b_{4}] + ⋯ + [b_{n} - b_{n + 1}] + ⋯$
is a telescoping series. The $k th$ partial sum of this series is given by $S_{k} = b_{1} - b_{k + 1} .$ The series will converge if and only if $lim_{k \to \infty} b_{k + 1}$ exists. In that case,

$\sum_{n = 1}^{\infty} [b_{n} - b_{n + 1}] = b_{1} - lim_{k \to \infty} (b_{k + 1}) .$

If $lim_{n \to \infty} a_{n} \neq 0,$ then the series $\sum_{n = 1}^{\infty} a_{n}$ diverges.
If $lim_{n \to \infty} a_{n} = 0,$ the series $\sum_{n = 1}^{\infty} a_{n}$ may converge or diverge.
If $\sum_{n = 1}^{\infty} a_{n}$ is a series with positive terms $a_{n}$ and $f$ is a continuous, decreasing function such that $f (n) = a_{n}$ for all positive integers $n,$ then

$\sum_{n = 1}^{\infty} a_{n} and \int_{1}^{\infty} f (x) d x$

either both converge or both diverge. Furthermore, if $\sum_{n = 1}^{\infty} a_{n}$ converges, then the $N th$ partial sum approximation $S_{N}$ is accurate up to an error $R_{N}$ where $\int_{N + 1}^{\infty} f (x) d x < R_{N} < \int_{N}^{\infty} f (x) d x .$
The p-series $\sum_{n = 1}^{\infty} 1 / n^{p}$ converges if $p > 1$ and diverges if $p \leq 1 .$

The comparison tests are used to determine convergence or divergence of series with positive terms.
When using the comparison tests, a series $\sum_{n = 1}^{\infty} a_{n}$ is often compared to a geometric or p-series.

For an alternating series $\sum_{n = 1}^{\infty} {(−1)}^{n + 1} b_{n},$ if $b_{k + 1} \leq b_{k}$ for all $k$ and $b_{k} \to 0$ as $k \to \infty,$ the alternating series converges.
If $\sum_{n = 1}^{\infty} | a_{n} |$ converges, then $\sum_{n = 1}^{\infty} a_{n}$ converges.

For the ratio test, we consider

$ρ = lim_{n \to \infty} | \frac{a_{n + 1}}{a_{n}} | .$

If $ρ < 1,$ the series $\sum_{n = 1}^{\infty} a_{n}$ converges absolutely. If $ρ > 1,$ the series diverges. If $ρ = 1,$ the test does not provide any information. This test is useful for series whose terms involve factorials.
For the root test, we consider

$ρ = lim_{n \to \infty} \sqrt[n]{| a_{n} |} .$

If $ρ < 1,$ the series $\sum_{n = 1}^{\infty} a_{n}$ converges absolutely. If $ρ > 1,$ the series diverges. If $ρ = 1,$ the test does not provide any information. The root test is useful for series whose terms involve powers.
For a series that is similar to a geometric series or $p - series,$ consider one of the comparison tests.