\( s_{n}=\sum \limits_{k=3}^{n}\left(\frac{2}{5}\right)^{k} \) n≥3
\( s_{n}=\sum \limits_{k=0}^{n}\left(\frac{2}{5}\right)^{k} -\left(\frac{2}{5}\right)^{2}-\left(\frac{2}{5}\right)^{1}-\left(\frac{2}{5}\right)^{0} \)
\(=\frac{(\frac{2}{5})^{n+1}-1 }{\frac{2}{5}-1 } -\left(\frac{2}{5}\right)^{2}-\left(\frac{2}{5}\right)^{1}-\left(\frac{2}{5}\right)^{0} \)
\(=\frac{(\frac{2}{5})^{n+1}-1 }{-\frac{3}{5} } -\frac{4}{25}-\frac{2}{5}-1 \)
Für n gegen ∞ ist der Grenzwert
\(s=\frac{-1 }{-\frac{3}{5} } -\frac{4}{25}-\frac{2}{5}-1=\frac{5}{3} -\frac{4}{25}-\frac{2}{5}-1 \)
\( =\frac{125}{75} -\frac{12}{75}-\frac{30}{75}-1=\frac{8}{75} \)