Saved Bookmarks
| 1. |
Find the sum to infinite terms of the series `tan^(-1)((1)/(3))+tan^(-1)((2)/(9))+"........."+tan^(-1)((2^(n-1))/(1+2^(2n-1)))+"......"`. |
|
Answer» `T_(n)=tan^(-1)((2^(n-1))/(1+2^(2n-1)))=tan^(-1)((2^(n-1))/(1+2^(n)*2^(n-1)))` `=tan^(-1)((2^(n)-2^(n-1))/(1+2^(n)*2^(n-1)))=tan^(-1)2^(n)-tan^(-1)2^(n-1)` `S_(n)=T_(1)+T_(2)+"......."+T_(n)` `=(tan^(-1)2^(1)-tan^(-1)2^(0))+(tan^(-1)2^(2)-tan^(-1)2^(1))+"........"+(tan^(-1)2^(n)-tan^(-1)2^(n-1))` `=(tan^(-1)2^(n)-tan^(-1)1)` `S_(n)=tan^(-1)2^(n)-(pi)/(4)` `S=lim_(n to oo)S_(n)=lim_(n to oo)(tan^(-1)2^(n)-(pi)/(4))=(pi)/(2)-(pi)/(4)=(pi)/(4)`. |
|