Calculus Matching: Series, Limits & Fourier Coefficients

This solution provides step-by-step calculations to match entries from List-1 (P, Q, R) with appropriate values from List-2 (I, II, III).

P: Infinite Series Summation

We need to calculate the sum $ S = \sum_{n=1}^\infty \frac{1}{(n+2)(n+1)} $.

1. Use partial fractions: $ \frac{1}{(n+1)(n+2)} = \frac{1}{n+1} - \frac{1}{n+2} $.
2. Recognize the telescoping nature of the series: $ \sum_{n=1}^N \left( \frac{1}{n+1} - \frac{1}{n+2} \right) = \left( \frac{1}{2} - \frac{1}{3} \right) + \left( \frac{1}{3} - \frac{1}{4} \right) + \dots + \left( \frac{1}{N+1} - \frac{1}{N+2} \right) = \frac{1}{2} - \frac{1}{N+2} $
3. Find the limit as $N \to \infty$: $ S = \lim_{N \to \infty} \left( \frac{1}{2} - \frac{1}{N+2} \right) = \frac{1}{2} $

Result: P matches with III ($\frac{1}{2}$).

Q: Limit Evaluation using L'Hôpital's Rule

Evaluate the limit $ L = \lim_{x \to 0} \left( \frac{3}{x^2} \int_0^x \sin(t) dt \right) $.

1. Integral evaluation: $ \int_0^x \sin(t) dt = [-\cos(t)]_0^x = 1 - \cos(x) $.
2. Substitute into the limit: $ L = \lim_{x \to 0} \frac{3(1 - \cos x)}{x^2} $.
3. This is the indeterminate form $\frac{0}{0}$. Apply L'Hôpital's Rule: $ L = \lim_{x \to 0} \frac{3 \sin x}{2x} $.
4. The form is still $\frac{0}{0}$. Apply L'Hôpital's Rule again: $ L = \lim_{x \to 0} \frac{3 \cos x}{2} $.
5. Evaluate the final limit: $ L = \frac{3 \cos(0)}{2} = \frac{3}{2} $.

Result: Q matches with I ($\frac{3}{2}$).

R: Fourier Series Coefficient Sum

For the function $f(x) = \frac{1}{2} \sin x - \frac{1}{2} \cos x + \frac{1}{\sqrt{2}} \sin 2x$, find the sum $ \sum_{n=0}^\infty (a_n^2 + b_n^2) $.

Comparing $f(x)$ to the standard Fourier series $f(x) \sim \frac{a_0}{2} + \sum_{n=1}^\infty (a_n \cos nx + b_n \sin nx)$, we identify the coefficients:

• $a_0 = 0$
• $a_1 = -\frac{1}{2}$
• $b_1 = \frac{1}{2}$
• $a_2 = 0$
• $b_2 = \frac{1}{\sqrt{2}}$
• All other coefficients ($a_n, b_n$ for $n \ge 3$) are 0.

Calculate the sum $ \sum_{n=0}^\infty (a_n^2 + b_n^2) = a_0^2 + \sum_{n=1}^\infty (a_n^2 + b_n^2) $:

Result: R matches with II ($1$).

Final Matches Summary

The correct pairings derived from the calculations are:

• P $\longrightarrow$ III
• Q $\longrightarrow$ I
• R $\longrightarrow$ II

This corresponds to the option P $\longrightarrow$ III, Q $\longrightarrow$ I, R $\longrightarrow$ II.