Infi-2 10

Absolute and conditional convergence #definition

\begin{matrix} \int f is called absolutely convergent if \int | f | converges \\ \int f is called conditionally convergent if \int f converges but \int | f | diverges \end{matrix}

Absolute convergence preserves "regular" convergence #lemma

\begin{matrix} If \int | f | converges, then \int f also converges \\ Proof: \\ Let \int | f | converges \\ Let f_{+} = {\begin{matrix} f & f \geq 0 \\ 0 & f < 0 \end{matrix} \\ Let f_{-} = {\begin{matrix} 0 & f \geq 0 \\ - f & f < 0 \end{matrix} \\ 0 \leq f_{+} \leq | f | ⟹ \int f_{+} converges \\ 0 \leq f_{-} \leq | f | ⟹ \int f_{-} converges \\ \int f = \int (f_{+} - f_{-}) = \underset{Converges}{\underset{⏟}{\int f_{+}}} - \underset{Converges}{\underset{⏟}{\int f_{-}}} ⟹ \int f converges \end{matrix}

Dirichlet's convergence test for integrals #theorem

\begin{matrix} Let f be a continuously differentiable, monotonically decreasing function \\ lim_{x \to \infty} f (x) = 0 \\ Let g be continuous \\ Let G (x) = \int_{a}^{x} g (t) d t be bounded \\ Then \int_{a}^{\infty} f (x) g (x) d x converges \\ Proof: \\ Let f be a continuously differentiable, monotonically decreasing function \\ lim_{x \to \infty} f (x) = 0 \\ Let g be continuous \\ Let G (x) = \int_{a}^{x} g (t) d t be bounded \\ \int_{a}^{\infty} f (x) g (x) d x = lim_{b \to \infty} \int_{a}^{b} f (x) g (x) d x \\ \int_{a}^{b} f (x) g (x) d x = \int_{a}^{b} f (x) G^{'} (x) d x = f (x) G (x) |_{x = a}^{x = b} - \int_{a}^{b} f^{'} (x) G (x) d x \\ f (x) G (x) |_{x = a}^{x = b} = \underset{\to 0}{\underset{⏟}{f (b)}} \underset{Bounded}{\underset{⏟}{G (b)}} - f (a) \underset{\int_{a}^{a} = 0}{\underset{⏟}{G (a)}} \underset{b \to \infty}{\to} 0 \\ It is now enough to show that \int_{a}^{\infty} f^{'} (x) G (x) d x converges \\ Consequently, we can just show that \int_{a}^{\infty} | f^{'} (x) G (x) | d x converges \\ | G (x) | \leq M ⟹ \int_{a}^{\infty} | f^{'} (x) G (x) | d x \leq M \int_{a}^{\infty} | f^{'} (x) | d x \\ f is monotonically decreasing ⟹ f^{'} (x) < 0 \\ ⟹ M \int_{a}^{\infty} | f^{'} (x) | d x = - M \int_{a}^{\infty} f^{'} (x) d x = - M lim_{b \to \infty} \int_{a}^{b} f^{'} (x) d x = - M lim_{b \to \infty} f (x) |_{x = a}^{x = b} = \\ = - M lim_{b \to \infty} (\underset{\to 0}{\underset{⏟}{f (b)}} - f (a)) = M f (a) \\ ⟹ \int_{a}^{\infty} | f^{'} (x) G (x) | converges \\ ⟹ \int_{a}^{\infty} f (x) g (x) d x converges \end{matrix}