The value of `tan^-1(1/3)+tan^-1(2/9)+tan^-1(4/33)+tan^-1(8/129)+....n` terms is:

Explanation

`T_{r}=tan ^{-1}(frac{2^{r-1}}{2^{2 r-1}+1})`
`
r_{1}=frac{1}{3}`
`
r_{2}=frac{2}{9}`
`
r_{3}=frac{4}{33}`
`
2^{r-1}=2 cdot 2^{r-1}-2^{r-1}=2^{r}-2^{r-1}`
`
T_{r}=tan ^{-1}(frac{2^{r} cdot 2^{r-1}}{1+2^{r} cdot 2^{r-1}})`
`
T_{1}=tan ^{-1}(2)-tan ^{1}(1) `
`
T_{2}=tan ^{-1}(4)-tan ^{-1}(2) `
`
...