Dobiński, parzyste i nie
: 20 lip 2024, o 05:00
\(\displaystyle{ \frac{1}{e} \sum_{n=0}^{ \infty } \frac{\left( 2n+1\right) ^{k+1} }{\left( 2n+1\right)! } = \frac{1}{2e} \sum_{n=0}^{ \infty } \frac{n ^{k+1} +\left( -1\right) ^{n} \left( n+1\right) ^{k} }{n!} }\)
\(\displaystyle{ \frac{1}{e} \sum_{n=0}^{ \infty } \frac{\left( 2n\right) ^{k+1} }{\left( 2n\right)! } = \frac{1}{2e} \sum_{n=0}^{ \infty } \frac{n ^{k+1} -\left( -1\right) ^{n} \left( n+1\right) ^{k} }{n!} }\)
\(\displaystyle{ \frac{1}{e} \sum_{n=0}^{ \infty } \frac{\left( 2n\right) ^{k+1} }{\left( 2n\right)! } = \frac{1}{2e} \sum_{n=0}^{ \infty } \frac{n ^{k+1} -\left( -1\right) ^{n} \left( n+1\right) ^{k} }{n!} }\)