\(\displaystyle{ \sum_{n=1}^{ \infty } \frac{1}{k ^{p _{n} } } = \frac{k ^{7} + k ^{6} + k ^{4} + k ^{2} - k - 1 }{k ^{9} - k ^{3} } ,k \ge 2 }\)
\(\displaystyle{ \sum_{n=1}^{ \infty } \frac{\left( -1\right) ^{n+1} }{k ^{p _{n} } } = \frac{k ^{5} - k ^{4} + k ^{3} + k - 1 }{k ^{7} + k ^{5} + k ^{3} } ,k \ge 2 }\)
\(\displaystyle{ p _{n} \hbox{ - n-ta liczba pierwsza} }\)
Sumy z pierwszymi w wykładniku
Re: Sumy z pierwszymi w wykładniku
\(\displaystyle{ \sum_{n=1}^{ \infty } \frac{1}{k ^{p _{n}+n } } = \frac{k ^{3}+k-1 }{k ^{6} - k ^{3} } ,k \ge 2}\)
\(\displaystyle{ \sum_{n=1}^{ \infty } \frac{\left( -1\right) ^{n+1} }{k ^{p _{n}+n } } = \frac{k ^{3}-k+1 }{k ^{6} + k ^{3} } ,k \ge 2}\)
\(\displaystyle{ \sum_{n=1}^{ \infty } \frac{1}{k ^{p _{n}-n } } = \frac{2k ^{3}-k ^{2}+2k-2}{k ^{4}-k ^{3}+k ^{2}-k } ,k \ge 2 }\)
\(\displaystyle{ \sum_{n=1}^{ \infty } \frac{\left( -1\right) ^{n+1} }{k ^{p _{n}-n } } = \frac{k }{k ^{3}+k ^{2}+k+1 } ,k \ge 2 }\)
\(\displaystyle{ \sum_{n=1}^{ \infty } \frac{\left( -1\right) ^{n+1} }{k ^{p _{n}+n } } = \frac{k ^{3}-k+1 }{k ^{6} + k ^{3} } ,k \ge 2}\)
\(\displaystyle{ \sum_{n=1}^{ \infty } \frac{1}{k ^{p _{n}-n } } = \frac{2k ^{3}-k ^{2}+2k-2}{k ^{4}-k ^{3}+k ^{2}-k } ,k \ge 2 }\)
\(\displaystyle{ \sum_{n=1}^{ \infty } \frac{\left( -1\right) ^{n+1} }{k ^{p _{n}-n } } = \frac{k }{k ^{3}+k ^{2}+k+1 } ,k \ge 2 }\)
Re: Sumy z pierwszymi w wykładniku
\(\displaystyle{ \sum_{n=1}^{ \infty }k ^{-p _{2n-1} } = \frac{k ^{7} + k ^{4} - k }{k ^{9} - k ^{3} } ,k \ge 2 }\)
\(\displaystyle{ \sum_{n=1}^{ \infty }k ^{-p _{2n} } = \frac{k ^{6} + k ^{2} - 1 }{k ^{9} - k ^{3} } ,k \ge 2 }\)
\(\displaystyle{ \sum_{n=1}^{ \infty }k ^{-p _{2n} } = \frac{k ^{6} + k ^{2} - 1 }{k ^{9} - k ^{3} } ,k \ge 2 }\)
Re: Sumy z pierwszymi w wykładniku
\(\displaystyle{ \sum_{n=1}^{ \infty } \left( -1\right) ^{n+1}k ^{-p _{2n-1} }+ \left( -1\right) ^{n+1}k ^{-p _{2n} } = \frac{k ^{7}+k ^{6}-k ^{4}-k ^{2}+k+1 }{k ^{9}+k ^{3}},k \ge 2 }\)
\(\displaystyle{ \sum_{n=1}^{ \infty } \left( -1\right) ^{n+1}k ^{-p _{2n-1} }=\frac{k ^{7}-k ^{4}+k}{k ^{9}+k ^{3}},k \ge 2 }\)
\(\displaystyle{ \sum_{n=1}^{ \infty } \left( -1\right) ^{n+1}k ^{-p _{2n} }=\frac{k ^{6}-k ^{2}+1}{k ^{9}+k ^{3}},k \ge 2 }\)
\(\displaystyle{ \sum_{n=1}^{ \infty } \left( -1\right) ^{n+1}k ^{-p _{2n-1} }=\frac{k ^{7}-k ^{4}+k}{k ^{9}+k ^{3}},k \ge 2 }\)
\(\displaystyle{ \sum_{n=1}^{ \infty } \left( -1\right) ^{n+1}k ^{-p _{2n} }=\frac{k ^{6}-k ^{2}+1}{k ^{9}+k ^{3}},k \ge 2 }\)