Question and Answers Forum

All Questions      Topic List

Trigonometry Questions

Previous in All Question      Next in All Question      

Previous in Trigonometry      Next in Trigonometry      

Question Number 30929 by rahul 19 last updated on 28/Feb/18

$$\underset{n=0}{\sum ^{\mathrm{\infty }}}{\tan }^{-1}(n+\frac{1}{2})-{\tan }^{-1}(n-\frac{1}{2})=?$$

Commented by abdo imad last updated on 01/Mar/18

$$letput{S}_n=\sum _{k=0}^{n}arctan(k+\frac{1}{2})-arctan(k-\frac{1}{2})$$$$S_n=\sum _{k=0}^{n}arctan\left(\frac{2k+1}{2}\right)-actan\left(\frac{2k-1}{2}\right)$$$$=\sum _{k=0}^n(v_k-v_{k-1})with{v}_k=arctan\left(\frac{2k+1}{2}\right)$$$$S_n=v_0-v_{-1}+v_1-v_o+...+v_n-v_{n-1}=v_n-v_{-1}\Rightarrow$$$$S_n=arctan\left(\frac{2n+1}{2}\right)-arctan(-\frac{1}{2})\Rightarrow$$$${lim}_{n\to \mathrm{\infty }}{S}_n=\frac{\pi }{2}+arctan\left(\frac{1}{2}\right)=\frac{\pi }{2}+\frac{\pi }{2}-arctan2$$$$=\pi -arctan2.$$

Answered by sma3l2996 last updated on 28/Feb/18

$$A=\underset{n=0}{\sum ^{\mathrm{\infty }}}{tan}^{-1}\left(\frac{2n+1}{2}\right)-\underset{n=0}{\sum ^{\mathrm{\infty }}}{tan}^{-1}\left(\frac{2n-1}{2}\right)$$$$letk=2n+1andi=2n-1$$$$soA=\underset{k=1}{\sum ^{\mathrm{\infty }}}{tan}^{-1}\left(k/2\right)-\underset{i=-1}{\sum ^{\mathrm{\infty }}}{tan}^{-1}\left(i/2\right)$$$$=\underset{k=1}{\sum ^{\mathrm{\infty }}}{tan}^{-1}\left(k/2\right)-\underset{i=1}{\sum ^{\mathrm{\infty }}}{tan}^{-1}\left(i/2\right)-{tan}^{-1}(-1/2)-{tan}^{-1}\left(0\right)$$$$=-(-\pi )=\pi$$

Answered by ajfour last updated on 28/Feb/18

$$={\tan }^{-1}\left(\frac{1}{2}\right)-{\tan }^{-1}\left(\frac{-1}{2}\right)+$$$${\tan }^{-1}\left(\frac{3}{2}\right)-{\tan }^{-1}\left(\frac{1}{2}\right)+$$$$...........................+$$$$...........................+$$$$+{\tan }^{-1}(N+\frac{1}{2})-{\tan }^{-1}(N-\frac{1}{2})$$$$={\tan }^{-1}(N+\frac{1}{2})-{\tan }^{-1}\left(\frac{-1}{2}\right)$$$$withN\to \mathrm{\infty }$$$$=\frac{\pi }{2}+{\tan }^{-1}\left(\frac{1}{2}\right).$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com