Infi-2 11

Convergence of integral and series (Integral convergence test) #theorem

\begin{matrix} Let f be a continuous, monotonically decreasing and positive function on [a, \infty) \\ Then \int_{a}^{\infty} f (x) d x converges ⟺ \sum_{n = a}^{\infty} f (n) converges \\ Note: \\ Integral and series might not converge to the same limit(!) \\ \sum_{n = 1}^{\infty} \frac{1}{n^{2}} = \frac{π^{2}}{6} but \int_{1}^{\infty} \frac{1}{x^{2}} d x = 1 \\ \sum_{n = 0}^{\infty} e^{- n} = \frac{1}{1 - \frac{1}{e}} but \int_{0}^{\infty} e^{- x} d x = 1 \end{matrix}

\begin{matrix} \int_{7}^{\infty} {(1 - \frac{1}{x})}^{x^{2}} d x ⟺ \sum_{n = 7}^{\infty} {(1 - \frac{1}{n})}^{n^{2}} ⟺ lim_{n \to \infty} \sqrt[n]{{(1 - \frac{1}{n})}^{n^{2}}} < 1 \\ ⟺ lim_{n \to \infty} {(1 - \frac{1}{n})}^{n} < 1 ⟺ e^{- 1} < 1 \end{matrix}

\begin{matrix} \int_{a}^{\infty} f (x) d x converges \overset{?}{⟹} lim_{x \to \infty} f (x) = 0 \\ No \\ \int_{a}^{\infty} \sin (x^{6}) d x converges, but lim_{x \to \infty} \sin (x^{6}) \neq 0 \end{matrix}