Differential Calculus
Differential Calculus
star_batch_jee_advanced_2025
Grade 12
Question:
If $a^2 + b^2 = 1$ and $u$ is the minimum value of $\frac{b+1}{a+b-2}$, then find the value of $u^2$.
Step-by-Step Solution
Key Concept: Recognizing a harmonic sum as a Riemann sum to evaluate its asymptotic behavior using integration.
Given $H_n = \frac{1}{n+1} + \frac{1}{n+2} + ... + \frac{1}{n+n}$, we find $\frac{n}{H_n} = \lim_{n \to \infty} \sum_{r=1}^{n} \frac{n}{n+r}$ which can be rewritten as a Riemann sum: $\lim_{n \to \infty} \sum_{r=1}^{n} \frac{1}{1+r/n} = \int_0^1 \frac{dx}{1+x} = \ln 2$. Therefore $\lim_{n \to \infty} \frac{H_n}{n} = \frac{1}{\ln 2}$.
Correct Answer: 9