If $\int_0^{\pi / 2} \tan ^{14}\left(\frac{x}{2}\right) d x=2\left[\sum_{n=1}^7 f(n)-\frac{\pi}{4}\right]$, then $f(n)=$

• A

  $\frac{(-1)^n}{n-1}$
• B

  $\frac{(-1)^n}{2 n+1}$
• Correct answer
  C

  $\frac{(-1)^{n+1}}{2 n-1}$
• D

  $\frac{(-1)^{n+1}}{n+1}$

Hence, the correct answer is option: 'C'.