For any positive integer n, define fn:(0,∞)→Ras fn(x)=j=1∑ntan−1(1+(x+j)(x+j−1)1)for all x∈(0,∞).Here, the inverse trigonometric function tan−1xassumes values in (−2π,2π)˙Then, which of the following statement(s) is (are) TRUE?j=1∑5tan2(fj(0))=55(b) j=1∑10(1+fj′(0))sec2(fj(0))=10(c) For any fixed positive integer n, (lim)x→∞tan(fn(x))=n1(d) For any fixed positive integer n, (lim)x→∞sec2(fn(x))=1

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For any positive integer {n}, define {f}_{{n}}:{({0},)}{R}as {

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For any positive integer $n$ , define $f_{n} : (0, \infty) \to R$ as $f_{n} (x) = j = 1 \sum n tan^{- 1} (\frac{1}{1 + ( x + j ) ( x + j - 1 )})$ for all $x \in (0, \infty)$ .Here, the inverse trigonometric function $tan^{- 1} x$ assumes values in $(- \frac{π}{2}, \frac{π}{2}) ˙$ Then, which of the following statement(s) is (are) TRUE? $j = 1 \sum 5 tan^{2} (f_{j} (0)) = 55$ (b) $j = 1 \sum 10 (1 + f j^{'} (0)) sec^{2} (f_{j} (0)) = 10$ (c) For any fixed positive integer $n$ , $(lim)_{x \to \infty} tan (f_{n} (x)) = \frac{1}{n}$ (d) For any fixed positive integer $n$ , $(lim)_{x \to \infty} sec^{2} (f_{n} (x)) = 1$

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Michael Discussed

For any positive integer

n

, define

f_{n} : (0, \infty) \to R

f_{n} (x) = j = 1 \sum n tan^{- 1} (\frac{1}{1 + ( x + j ) ( x + j - 1 )})

for all

x \in (0, \infty)

.Here, the inverse trigonometric function

tan^{- 1} x

assumes values in

(- \frac{π}{2}, \frac{π}{2}) ˙

Then, which of the following statement(s) is (are) TRUE?

j = 1 \sum 5 tan^{2} (f_{j} (0)) = 55

(b)

j = 1 \sum 10 (1 + f j^{'} (0)) sec^{2} (f_{j} (0)) = 10

n

(lim)_{x \to \infty} tan (f_{n} (x)) = \frac{1}{n}

(d) For any fixed positive integer

n

(lim)_{x \to \infty} sec^{2} (f_{n} (x)) = 1

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Question Text	For any positive integer $n$ , define $f_{n} : (0, \infty) \to R$ as $f_{n} (x) = j = 1 \sum n tan^{- 1} (\frac{1}{1 + ( x + j ) ( x + j - 1 )})$ for all $x \in (0, \infty)$ .Here, the inverse trigonometric function $tan^{- 1} x$ assumes values in $(- \frac{π}{2}, \frac{π}{2}) ˙$ Then, which of the following statement(s) is (are) TRUE? $j = 1 \sum 5 tan^{2} (f_{j} (0)) = 55$ (b) $j = 1 \sum 10 (1 + f j^{'} (0)) sec^{2} (f_{j} (0)) = 10$ (c) For any fixed positive integer $n$ , $(lim)_{x \to \infty} tan (f_{n} (x)) = \frac{1}{n}$ (d) For any fixed positive integer $n$ , $(lim)_{x \to \infty} sec^{2} (f_{n} (x)) = 1$
Answer Type	Video solution: 1
Upvotes	150