Infi-1 15

Bounded series #definition

\begin{matrix} Series are called bounded if the sequence of it’s partial sums is bounded \\ \exists M : \forall N \in N : | \sum_{n = 1}^{N} | \leq M \end{matrix}

Dirichlet's test #theorem

\begin{matrix} \sum b_{n} is bounded \\ a_{n} monotonically non-increasing, a_{n} \to 0 \\ Then \sum b_{n} a_{n} \to M_{1} \end{matrix}

\begin{matrix} Example: \\ \sum \frac{(- 1)^{n}}{n} = \sum (- 1)^{n} \cdot \frac{1}{n} \\ | \sum (- 1)^{n} | \leq 1 \\ \frac{1}{n} \to 0 \\ ⟹ \sum \frac{(- 1)^{n}}{n} \to M_{1} \end{matrix}

\begin{matrix} Proof: \\ a_{n} \to 0 and a_{n} is monotonically non-increasing \\ ⟹ \forall n \in N : a_{n} \geq 0 \\ \sum b_{n} is bounded ⟹ S_{N} = \sum_{n = 1}^{N} b_{n} is bounded \\ \sum a_{n} b_{n} \to L ⟺ \forall ε > 0 : \exists N_{ε} : \forall N, M > N_{ε} \in N : | \sum_{n = M + 1}^{N} a_{n} b_{n} | < ε \\ | \sum_{n = M + 1}^{N} a_{n} b_{n} | = | \sum_{n = M + 1}^{N} a_{n} (S_{n} - S_{n - 1}) | = | \sum_{n = M + 1}^{N} a_{n} S_{n} - \sum_{n = M + 1}^{N} a_{n} S_{n - 1} | = \\ = | - S_{M} a_{M + 1} + \sum_{n = M + 1}^{N - 1} S_{n} (a_{n} - a_{n + 1}) + S_{N} a_{N} | \leq \\ \leq | - S_{M} a_{M + 1} | + \sum_{n = M + 1}^{N - 1} | S_{n} (a_{n} - a_{n + 1}) | + | S_{N} a_{N} | = \\ = | S_{M} | \cdot | a_{M + 1} | + \sum_{n = M + 1}^{N - 1} | S_{n} | \cdot | (a_{n} - a_{n + 1}) | + | S_{N} | \cdot | a_{N} | \\ | S_{n} | \leq K \\ a_{N}, a_{M + 1} \geq 0 \\ a_{M + 1} \geq a_{M + 2} \geq \dots \\ ⟹ a_{M + 1} - a_{M + 2} \geq 0 \\ ⟹ | S_{M} | \cdot | a_{M + 1} | + \sum_{n = M + 1}^{N - 1} | S_{n} | \cdot | (a_{n} - a_{n + 1}) | + | S_{N} | \cdot | a_{N} | \leq \\ K \cdot (a_{M + 1} + \sum_{n = M + 1}^{N} (a_{n} - a_{n + 1}) + a_{N}) \\ ⟹ | \sum_{n = M + 1}^{N} a_{n} b_{n} | \leq K \cdot (a_{M + 1} + a_{M + 1} - a_{M + 2} + \dots + a_{N - 1} - a_{N} + a_{N}) = \\ = K \cdot 2 a_{M + 1} \\ | \sum_{n = M + 1}^{N} a_{n} b_{n} | \leq 2 K \cdot a_{M + 1} \\ \forall ε > 0 : \exists N_{ε} : \forall M > N_{ε} : | a_{M + 1} | < \frac{ε}{2 K} \\ ⟹ \forall ε > 0 : \exists N_{ε} : \forall M > N_{ε} : | \sum_{n = M + 1}^{N} a_{n} b_{n} | \leq 2 K \cdot a_{m + 1} < 2 K \cdot \frac{ε}{2 K} = ε \end{matrix}

\begin{matrix} \sum (- 1)^{n}, \sum \sin (n), \sum \cos (n) are bounded series \end{matrix}

\begin{matrix} \sum \sin (n) is bounded ⟹ \sum \frac{\sin (n)}{n} = \sum (\sin (n) \cdot \underset{\to 0}{\underset{⏟}{\frac{1}{n}}}) \to M_{1} \end{matrix}