Limits, Continuity & Differentiability
Limit of Sum as Riemann Integral
nta_pyq_2023_apr
Grade 12
Question:
Among (S1): $\displaystyle\lim_{n\to\infty}\dfrac{1}{n^2}(2+4+6+\cdots+2n)=1$ and (S2): $\displaystyle\lim_{n\to\infty}\dfrac{1}{n^{16}}(1^{15}+2^{15}+\cdots+n^{15})=\dfrac{1}{16}$,
Both (S1) and (S2) are true
Only (S1) is true
Both (S1) and (S2) are false
Only (S2) is true
Step-by-Step Solution
Key Concept: (S1): $\frac{2}{n^2}\cdot\frac{n(n+1)}{2}\to1$. (S2): Riemann sum $\int_0^1 x^{15}dx=\frac{1}{16}$.
Both are true.
Correct Answer: 1