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MMLU-Pro / 8856 / For all n>1, define …

Problem

Answer published by the source. Consult the official source to check your work against its answer.

question

For all n>1n>1, define an=∑k=1n−1sin((2k−1)π2n)cos2((k−1)π2n)cos2(kπ2n)a_n=\sum_{k=1}^{n-1} \frac{\sin (\frac{(2 k-1) \pi}{2 n})}{\cos ^2(\frac{(k-1) \pi}{2n}) \cos ^2 (\frac{k \pi}{2n})}. What is the limit of an/n3a_n/n^3 as nn goes to infinity?
Plain-text mathematical notation (without MathML)
For all n>1, define a_(n)=∑_(k=1)^(n−1)(sin(((2k−1)π)/(2n)))/(cos²(((k−1)π)/(2n))cos²((kπ)/(2n))). What is the limit of a_(n)/n³ as n goes to infinity?
Original LaTeX notation
For all $n>1$, define $a_n=\sum_{k=1}^{n-1} \frac{\sin (\frac{(2 k-1) \pi}{2 n})}{\cos ^2(\frac{(k-1) \pi}{2n}) \cos ^2 (\frac{k \pi}{2n})}$. What is the limit of $a_n/n^3$ as $n$ goes to infinity?

category

math

options

  1. 0.666
  2. 0.125
  3. 1.000
  4. 0.750
  5. 0.414
  6. 0.100
  7. 0.258
  8. 0.500
  9. 0.333
  10. 0.025

src

theoremQA-Math

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Source and history

upstream_id / Original problem ID
8856
task_id
aa791e52-73aa-5eff-a1f5-ffcb90a487d2
task_key
default--test--8856
task_revision_id
1
config / split
default / test

Official source

initial import