Limits, Continuity & Differentiability
Differential Calculus-1
star_batch_jee_advanced_2025
Grade 12
The value of $\lim_{n \to \infty} \left( \frac{1}{\sqrt{n^2}} + \frac{1}{\sqrt{n^2+1}} + \ldots + \frac{1}{\sqrt{n^2+2n}} \right)$ is ______.
Step-by-Step Solution
Key Concept: Squeeze theorem applied by bounding the sum between two expressions that converge to the same limit.
The function $f(x) = \frac{1}{\sqrt{n^2}} + \frac{1}{\sqrt{n^2+1}} + \frac{1}{\sqrt{n^2+2}} + \ldots + \frac{1}{\sqrt{n^2+2n}}$ is bounded between two Riemann sums. By observing that $\frac{1}{\sqrt{n^2+2n}} < \frac{1}{\sqrt{n^2+1}} < \frac{1}{\sqrt{n^2}}$, we establish that $2 < \lim_{n \to \infty} f(x) < 2$, which forces the limit to equal $2$.
Correct Answer: I need to find the limit of the sum:
$$\lim_{n \to \infty} \left( \frac{1}{\sqrt{n^2}} + \frac{1}{\sqrt{n^2+1}} + \ldots + \frac{1}{\sqrt{n^2+2n}} \right)$$