求級數\( \displaystyle \frac{1}{3 \times 7}+\frac{1}{7 \times 11}+\frac{1}{11 \times 15}+\frac{1}{15 \times 19}+\ldots+\frac{1}{(4n-1)\times(4n+3)}+\ldots \)，請各位大大幫忙給出算式，謝謝

忘了給答案，答案是 n/(3(4n+3))

裂項相消

\( \frac{1}{(4k-1)(4k+3)} = \frac{1}{4} \left( \frac{1}{4k-1} - \frac{1}{4k+3} \right) \)

故原式 \( = \frac{1}{4} \left( \frac{1}{3} - \frac17 + \frac17 -\frac1{11} + \ldots \frac{1}{4n-1} -\frac{1}{4n+3} \right) \)

\( = \frac{1}{4} \left( \frac{1}{3} - \frac{1}{4n+3} \right) = \frac{n}{3(4n+3)} \)

謝謝大大！