sum from n=1 to infinity of 1/(n(n+3))

解答

n = 1 \sum \infty \frac{1}{n ( n + 3 )}

\frac{11}{18}

十进制

0.61111 \dots

求解步骤

n = 1 \sum \infty \frac{1}{n ( n + 3 )}

将

\frac{1}{n ( n + 3 )}

用部份分式展开:

\frac{1}{3 n} - \frac{1}{3 ( n + 3 )}

= n = 1 \sum \infty \frac{1}{3 n} - \frac{1}{3 ( n + 3 )}

展开裂项级数:

\frac{11}{18}

= \frac{11}{18}

d er i v a t i v e y = x^{5}

x \to 3 lim (\frac{x + 2}{x - 2})

\frac{d}{d x} (cos (\frac{π}{3}))

d er i v a t i v e x^{2} + 10 x + c

in v erse l a pl a ce \frac{1}{s ^{3} ( s + 3 )}