find the sum of sigma from n =1 to infinity of [arctan(n+1)- arctann]
Hello, twilightstr!
We have:Find the sum of: .$\displaystyle S \;=\;\sum^{\infty}_{n=1}\bigg[\arctan(n+1)- \arctan(n)\bigg]$
. . $\displaystyle S \;=\;\bigg[{\color{red}\rlap{////////}}\arctan(2) - \arctan(1)\bigg] +$ $\displaystyle \bigg[{\color{green}\rlap{////////}}\arctan(3) - {\color{red}\rlap{////////}}\arctan(2)\bigg] +$ $\displaystyle \bigg[{\color{blue}\rlap{////////}}\arctan(4) - {\color{green}\rlap{////////}}\arctan(3)\bigg] + \hdots $
. . $\displaystyle S \;=\;-\arctan(1)$
. . $\displaystyle S\;=\;-\frac{\pi}{4}$
Hello, Spec!
Aren't you forgetting the term $\displaystyle \arctan \infty$ ?
$\displaystyle S\:=\:-\frac{\pi}{4}+\lim_{n \to \infty}\arctan (n+1)=-\frac{\pi}{4}+\frac{\pi}{2}=\frac{\pi}{4}$
Yes, I did!
I forgot to look at the other end of the series . . . *blush*