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Matematikai alapok GY
Analízis 1. GY
Analízis 2. GY
Analízis 3. GY

Analysis I. Pr.
Analysis II. Pr.
Analysis III. Pr.

1. $$ S_1 := \sum\limits_{n=1}^\infty \left(\frac{1}{2}\right)^n = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots $$ $$ 2S_1 = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots = S_1 + 1 $$ $$\boxed{S_1 = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots = 1} \qquad \color{green}{\huge ✓} $$ 2. $$ S_2 := \sum\limits_{n=1}^\infty (-1)^{n-1} = 1 - 1 + 1 - 1 + \dots $$ $$ \begin{array}{lrllllllllllll} & S_2 & = & 1 & - & 1 & + & 1 & - & 1 & + & \dots \\ \oplus & S_2 & = &&& 1 & - & 1 & + & 1 & - & \dots \\\hline & 2S_2 & = & 1 & + & 0 & + & 0 & + & 0 & + & \dots & = & 1 \end{array} $$ $$ \boxed{S_2 = 1 - 1 + 1 - 1 + \dots = \frac{1}{2}} \qquad \color{red}{\huge ✗} $$ 3. $$ S_3 := \sum\limits_{n=1}^\infty (-1)^{n-1} \cdot n = 1 - 2 + 3 - 4 + \dots $$ $$ \begin{array}{lrllllllllllll} & S_2 & = & 1 & - & 1 & + & 1 & - & 1 & + & \dots \\ \ominus & S_3 & = & 1 & - & 2 & + & 3 & - & 4 & + & \dots \\\hline & S_2-S_3 & = & 0 & + & 1 & - & 2 & + & 3 & - & \dots & = & S_3 \end{array} $$ $$ \frac{1}{2} - S_3 = S_3 $$ $$ \boxed{S_3 = 1 - 2 + 3 - 4 + \dots = \frac{1}{4}} \qquad \color{red}{\huge ✗} $$ 4. $$ S_4 := \sum\limits_{n=1}^\infty n = 1 + 2 + 3 + 4 + \dots $$ $$ \begin{array}{lrllllllllllll} & S_4 & = & 1 & + & 2 & + & 3 & + & 4 & + & \dots \\ \ominus & S_3 & = & 1 & - & 2 & + & 3 & - & 4 & + & \dots \\\hline & S_4-S_3 & = & 0 & + & 4 & + & 0 & + & 8 & + & \dots & = & 4S_4 \end{array} $$ $$ S_4 - \frac{1}{4} = 4S_4 $$ $$ \boxed{S_4 = 1 + 2 + 3 + 4 + \dots = -\frac{1}{12}} \qquad \color{red}{\huge ✗} $$