What is the sum of the infinite telescopic series $\sum_{n=1}^\infty \arctan (n+4)-\arctan(n+2))$? Trying to find the sum of the telescopic series. Can't seem to figure it out! Please Help!
$$\sum_{1}^{\infty}(\tan^{-1}(n+4)-\tan^{-1}(n+2))$$
I've tried looking at other examples, but they're not helping.

Hint.  It doesn't matter much what the function is, telescopic series always work much the same way.  For example if we have a finite sum,
$$\eqalign{\smash{\sum_{n=1}^N \bigl(f(n+4)-f(n+2)\bigr)}
  &=f(5)-f(3)+f(6)-f(4)\cr
  &\quad{}+f(7)-f(5)+f(8)-f(6)\cr
  &\quad{}+\cdots\cr
  &\quad{}+f(N+3)-f(N+1)+f(N+4)-f(N+2)\ .\cr}$$
Now if you look carefully you will see that many terms cancel and we get
$$\sum_{n=1}^N \bigl(f(n+4)-f(n+2)\bigr)
  =-f(3)-f(4)+f(N+3)+f(N+4)\ .$$
For your problem you need to replace $f$ by $\arctan$, then see what happens as $N\to\infty$.
Good luck!
