\input /home/pasky/school/fastex/lib.tex

\theorem{1}{nutn� podm�nka konvergence �ady}{
	Nech� \M{\sum_{n=1}^{\infty} a_{n}} konverguje. Pak \M{\lim_{n\to\infty} a_{n} = 0}.


	\scpart{Pozorov�n�}{
		Pro geometrickou �adu \M{\sum_{n=1}^{\infty} q^{n-1}} plat�:

		$$ \sum_{n=1}^{\infty} a_{n} \textbox{konverguje} \Longleftrightarrow \lim_{n\to\infty} a_{n} = 0 $$

		(Implikace v�ak neplat� pro v�echny �ady.)
	}


	\proof{
		Nech� \M{s = \sum_{n=1}^{\infty} a_{n}}, pak $s \in \real$.
		$$ \lim_{n\to\infty} s_{n} = s $$
		$$ \lim_{n\to\infty} s_{n+1} = s $$
		$$ 0 = s - s = \lim_{n\to\infty} s_{n+1} - \lim_{n\to\infty} s_{n} \becauseof{\ref{VOAL}}{=} \lim_{n\to\infty} (s_{n+1}-s_{n}) = \lim_{n\to\infty} a_{n+1} $$
		$$ \lim_{n\to\infty} a_{n} = 0 $$

		\qed
	}


	\scpart{Varov�n�}{
		Implikaci nelze obr�tit (tedy podm�nka nen� posta�uj�c�).
		$$ \lim_{n\to\infty} a_{n} = 0 \not\Rightarrow \sum_{n=1}^{\infty} a_{n} \textbox{konverguje} $$
	}


	\example{

		Harmonick� �ada \M{\sum_{n=1}^{\infty}{1\over{}n} = +\infty}:
		$$ s_{m} = 1 + {1\over{}2} + \cdots + {1\over{}m} $$
		$$ s_{2m} = 1 + {1\over{}2} + \cdots + {1\over{}m} + {1\over{}{m+1}} + \cdots + {1\over{}{2m}} $$
		$$ s_{2m} - s_{m} = {1\over{}{m+1}} + {1\over{}{m+2}} + \cdots + {1\over{}{2m}} $$
			       $$ \ge {1\over{}{2m}} + {1\over{}{2m}} + \cdots + {1\over{}{2m}} $$
			       $$ = m{1\over{}{2m}} = {1\over{}2} $$
		$$ s_{2m} - s_{m} \ge {1\over{}2}\qquad \forall m\in\nat $$
		Tedy neplat� \reference{Bolzano-Cauchyova podm�nka}:
		$$ \becauseof{\ias{2}{12}{}}{\ge} \lim_{n\to\infty} s_{m} = +\infty $$



	} %: \example

}

\bend
