Question.
\(\lim_{n\rightarrow \infty }\frac{1}{n}\left [ \sqrt{\frac{1}{2}}+\frac{2}{n} +….+\frac{n}{n}\right ]\)=
(A)\(\frac{1}{2}\int_{0}^{1}\frac{1}{\sqrt{x}}dx\) (B)\(\int_{0}^{1}\sqrt{x}dx\) (C)\(\int_{0}^{1}xdx\) (D) \(\int_{1}^{2}xdx\) (E)\(2\int_{1}^{2}x\sqrt{x}dx\)
▶️ Answer/Explanation
Solution
1. Recognize the expression as a Riemann sum:
The given limit can be written as: \(\lim_{n\rightarrow \infty }\sum_{k=1}^{n} \sqrt{\frac{k}{n}} \cdot \frac{1}{n}\)
2. Identify the components:
• Function: \( f(x) = \sqrt{x} \)
• Interval: [0,1] divided into n equal subintervals
• Sample points: right endpoints \( x_k = \frac{k}{n} \)
• Interval: [0,1] divided into n equal subintervals
• Sample points: right endpoints \( x_k = \frac{k}{n} \)
3. The limit of a right Riemann sum equals the definite integral:
\(\lim_{n\rightarrow \infty }\sum_{k=1}^{n} f\left(\frac{k}{n}\right) \cdot \frac{1}{n} = \int_{0}^{1} \sqrt{x} \, dx\)
4. Verify against options:
The correct integral representation is \(\int_{0}^{1}\sqrt{x}dx\)
✅ Answer: B) \(\int_{0}^{1}\sqrt{x}dx\)
Note: This is a standard Riemann sum to integral conversion, where the sum represents the area under \( y = \sqrt{x} \) from 0 to 1 using right endpoints.
▶️ Answer/Explanation
Solution
1. Given function: \( f(x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}x^{2n-1}}{2n-1} \)
2. Differentiate term-by-term within the radius of convergence:
\( f'(x) = \sum_{n=1}^{\infty} \frac{d}{dx}\left( \frac{(-1)^{n+1}x^{2n-1}}{2n-1} \right) \)
3. Apply power rule to each term:
\( \frac{d}{dx}\left( \frac{(-1)^{n+1}x^{2n-1}}{2n-1} \right) = (-1)^{n+1}x^{2n-2} \)
4. Therefore, the derivative is:
\( f'(x) = \sum_{n=1}^{\infty} (-1)^{n+1}x^{2n-2} \)
5. Verify pattern:
For n=1: \( (-1)^2x^0 = 1 \)
For n=2: \( (-1)^3x^2 = -x^2 \)
For n=3: \( (-1)^4x^4 = x^4 \)
And so on…
For n=2: \( (-1)^3x^2 = -x^2 \)
For n=3: \( (-1)^4x^4 = x^4 \)
And so on…
✅ Answer: A) \(\sum_{n=1}^{\infty}(-1)^{n+1}x^{2n-2}\)
▶️ Answer/Explanation
Solution
1. Recognize the expression as a Riemann sum:
\(\frac{1}{n} \left[ \left( \frac{1}{n} \right)^2 + \left( \frac{2}{n} \right)^2 + \cdots + \left( \frac{3n}{n} \right)^2 \right] = \frac{1}{n} \sum_{k=1}^{3n} \left( \frac{k}{n} \right)^2\)
2. Identify the components:
• Function: \( f(x) = x^2 \)
• Interval: [0,3] divided into 3n subintervals of width \( \frac{1}{n} \)
• Sample points: right endpoints \( x_k = \frac{k}{n} \)
• Interval: [0,3] divided into 3n subintervals of width \( \frac{1}{n} \)
• Sample points: right endpoints \( x_k = \frac{k}{n} \)
3. The limit as n→∞ gives the definite integral:
\(\lim_{n\to\infty} \frac{1}{n} \sum_{k=1}^{3n} \left( \frac{k}{n} \right)^2 = \int_{0}^{3} x^2 dx\)
4. Verify against options:
The correct integral representation is \(\int_{0}^{3} x^2 dx\)
✅ Answer: D) \(\int_{0}^{3}x^{2}dx\)
Note: The expression represents a right Riemann sum for \( f(x) = x^2 \) over [0,3], which converges to the integral as n→∞. Options A, B, and C involve incorrect integrands, while option E incorrectly includes an extra factor of 3.
▶️ Answer/Explanation
Solution
1. Recognize the limit as a Riemann sum:
The expression \( \lim_{n\rightarrow \infty} \sum_{i=1}^{n} \sqrt{x_i} \Delta x \) represents the integral of \( \sqrt{x} \) from a to b.
2. Convert the Riemann sum to its corresponding integral:
\( \lim_{n\rightarrow \infty} \sum_{i=1}^{n} \sqrt{x_i} \Delta x = \int_{a}^{b} \sqrt{x} \, dx \)
3. Evaluate the integral:
\( \int \sqrt{x} \, dx = \frac{2}{3} x^{\frac{3}{2}} + C \)
Thus, \( \int_{a}^{b} \sqrt{x} \, dx = \frac{2}{3} \left( b^{\frac{3}{2}} – a^{\frac{3}{2}} \right) \)
4. Verify against options:
The correct answer matches option A.
✅ Answer: A) \( \frac{2}{3} \left( b^{\frac{3}{2}} – a^{\frac{3}{2}} \right) \)
Note: The Riemann sum represents the area under \( y = \sqrt{x} \) from a to b, which evaluates to \( \frac{2}{3} \) times the difference of the endpoint values raised to the 3/2 power.