szereg Maclaurina

[aatomka]

jak rozwinąć tę funkcję w szereg Maclaurina?

\(\displaystyle{ f(x) = \int_{0}^{x} \frac{arctg t}{t} dt}\)

[luka52]

\(\displaystyle{ \frac{\arctan t}{t} = \frac{1}{t} \sum_{n = 0}^{+\infty} \frac{(-1)^n}{2n+1} t^{2n + 1} = \sum_{n = 0}^{+\infty} \frac{(-1)^n}{2n+1} t^{2n}}\)
zatem

\(\displaystyle{ \int_0^x \frac{\arctan t}{t} \; \mbox d t = \sum_{n = 0}^{+\infty} \int_0^x \frac{(-1)^n}{2n+1} t^{2n} \; \mbox dt = \sum_{n = 0}^{+\infty}\frac{(-1)^n}{(2n+1)^2} x^{2n+1}}\)