alleryEvaluate $\displaystyle\lim_{n\to\infty} \dfrac{1^k + 2^k + \cdots + n^k}{n^{k+1}}$ for $k > -1$:
A$0$
B$\dfrac{1}{k+1}$
C$1$
D$k$
Answer & Solution
Correct answer: B. $\dfrac{1}{k+1}$
Recognise as Riemann sum: $\sum (r/n)^k \cdot (1/n) \to \int_0^1 x^k\, dx = 1/(k+1)$.
Keep practising Unit 6B: Definite Integration & Riemann Sums
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