Let $f_n(x) + f_n(y) = \frac{x^n + y^n}{x^n y^n}$ for all $x, y \in \mathbb{R} - \{0\}$ where $n \in \mathbb{N}$. Let $g(x) = \max\left\{f_2(x), f_3(x)\right\}$ for all $x \in \mathbb{R} - \{0\}$. The minimum value of $\sum_{k=1}^{\infty} f_{2k}(\csc \theta) + \sum_{k=1}^{\infty} f_{2k}(\sec \theta)$, where $\theta \neq \frac{k\pi}{2}; k \in \mathbb{I}$ is:

Sequences & Series

Infinite Series Summation

Grade 11

Question:

Let $f_n(x) + f_n(y) = \frac{x^n + y^n}{x^n y^n}$ for all $x, y \in \mathbb{R} - \{0\}$ where $n \in \mathbb{N}$.Let $g(x) = \max\left\{f_2(x), f_3(x)\right\}$ for all $x \in \mathbb{R} - \{0\}$.The minimum value of $\sum_{k=1}^{\infty} f_{2k}(\csc \theta) + \sum_{k=1}^{\infty} f_{2k}(\sec \theta)$, where $\theta \neq \frac{k\pi}{2}; k \in \mathbb{I}$ is:

(a) 1

(b) 2

(c) $\sqrt{2}$

(d) 4

Step-by-Step Solution

Key Concept: From the functional equation, determine $f_n(x) = x^{-n} + x^n$. Compute the infinite series using geometric series formula and minimize over $\theta$.

Answer: (b)

Correct Answer: B

Question:

Step-by-Step Solution

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