Consider two geometric progressions a1,a2,a3.......an & b1, b2, b3,.....bn with ar=1br=2r−1 and another sequence t1,t2,t3.......tn such that tr=cot−12ar+br then limn→∞ ∑r=1ntr is -

Question

Consider two geometric progressions

a_{1}, a_{2}, a_{3} ....... a_{n}

b_{1}, b_{2}, b_{3}, ..... b_{n}

with

a_{r} = \frac{1}{b_{r}} = 2^{r - 1}

and another sequence

t_{1}, t_{2}, t_{3} ....... t_{n}

such that

t_{r} = \cot^{- 1} (2 a_{r} + b_{r})

then

\lim_{n \to \infty} \sum_{r = 1}^{n} t_{r}

is -

0

π / 4

\tan^{- 1} 2

π / 2

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