Definite Integration
Limit of Sum
Grade 12
Question:
<p>Evaluate \(\displaystyle\lim_{n\to\infty}\frac{1^p+2^p+\cdots+n^p}{n^{p+1}}\), \(p>0\) [JEE Main 2014]</p>
<li>\(\dfrac{1}{p+1}\)</li>
<li>\(\dfrac{1}{p}\)</li>
<li>\(p+1\)</li>
<li>\(\dfrac{1}{p-1}\)</li>
Step-by-Step Solution
Key Concept: Riemann sum: (1/n) \cdot \Sigma(k/n)^p \to \int_0^1 x^p dx = 1/(p+1).
<div class='solution'>
<p>$$\frac{1^p+2^p+\cdots+n^p}{n^{p+1}}=\frac{1}{n}\sum_{k=1}^n\left(\frac{k}{n}\right)^p\to\int_0^1 x^p\,dx=\frac{1}{p+1}$$</p>
</div>
Correct Answer: A