# Thread: find the sum of a series q

1. ## find the sum of a series q

find the sum of sigma from n =1 to infinity of [arctan(n+1)- arctann]

2. $\displaystyle \sum_{n=1}^{\infty} \{\tan^{-1} (n+1) - \tan^{-1} (n) \} = \sum_{n=1}^ {\infty} \int_{n}^{n+1} \frac{dx}{1+x^{2}} = \int_{1}^{\infty} \frac{dx}{1+x^{2}} = \frac{\pi}{4}$

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

3. how did u get 1/(1-x)^2?

4. Remember that...

$\displaystyle \int \frac{dx}{1+x^{2}} = \tan^{-1} x + c$

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

5. Hello, twilightstr!

Find the sum of: .$\displaystyle S \;=\;\sum^{\infty}_{n=1}\bigg[\arctan(n+1)- \arctan(n)\bigg]$
We have:

. . $\displaystyle S \;=\;\bigg[{\color{red}\rlap{////////}}\arctan(2) - \arctan(1)\bigg] +$ $\displaystyle \bigg[{\color{green}\rlap{////////}}\arctan(3) - {\color{red}\rlap{////////}}\arctan(2)\bigg] +$ $\displaystyle \bigg[{\color{blue}\rlap{////////}}\arctan(4) - {\color{green}\rlap{////////}}\arctan(3)\bigg] + \hdots$

. . $\displaystyle S \;=\;-\arctan(1)$

. . $\displaystyle S\;=\;-\frac{\pi}{4}$

6. Aren't you forgetting the term $\displaystyle \arctan \infty$

$\displaystyle S=-\frac{\pi}{4}+\lim_{n \to \infty}\arctan (n+1)=-\frac{\pi}{4}+\frac{\pi}{2}=\frac{\pi}{4}$

7. 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*