darft-ce

$\sum a_{n}$ diverges, but $\sum a_{n} (a_{n} + 1)$ converges
- $a_{n} = - 1$
$\sum a_{n}$ converges, but $\sum a_{n} (a_{n} + 1)$ diverges
- $a_{n} = \frac{( - 1 ) ^{n}}{n}$
$\sum (a_{n} - a_{n + 1})$ converges, but $\sum a_{n}$ diverges
- $a_{n} = c$ for some constant $c \neq = 0$
$\sum a_{n}$ and $\sum b_{n}$ converge, but $\sum min (a_{n}, b_{n})$ diverges
- $a_{n} = \frac{( - 1 ) ^{n}}{n}$ , $b_{n} = \frac{( - 1 ) ^{n + 1}}{n}$
$n \to \infty lim n ∣ a_{n} ∣ = \frac{1}{c}$ but $\sum c^{n} a_{n}$ converges.
- $a_{n} = \frac{1}{c ^{n} n ^{2}}$

$\int_{a}^{\infty} f (x) d x$ converges, but $\int_{a}^{\infty} f (x) g (x) d x$ diverges ( $g$ is a continuous and bounded on $[a, \infty)$ )
- $f (x) = \frac{s i n x}{x}$ , $g (x) = sin x$

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