Liczba e po przecinku - inny szereg
: 14 maja 2024, o 20:48
\(\displaystyle{ e = 2 + \sum_{n=0}^{ \infty } \frac{1}{n!\left( n+3\right) } }\)
\(\displaystyle{ \displaystyle\sum_{n=0}^{ \infty } \dfrac{x^{n+2}}{n! }=x^2\displaystyle\sum_{n=0}^{ \infty } \dfrac{x^{n}}{n! }=x^2e^x}\)
\(\displaystyle{ \displaystyle\int_{0}^{1}\sum_{n=0}^{ \infty } \frac{x^{n+2}}{n! }dx=
\displaystyle\int_{0}^{1}x^2e^x}\)
\(\displaystyle{ \displaystyle\sum_{n=0}^{ \infty } \dfrac{1}{(n+3)n! }dx=
\displaystyle\int_{0}^{1}x^2e^x=e-2}\)