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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2020 • Question 8

Question

Math 6010 • Linear Models Spring 2020 • Question 8

Use your limit theorems to prove that the following limit exists and to compute its value: \[ \lim _{n\to \infty }\int _{-\sqrt {n}}^{\sqrt {n}} \left (1-\frac {x^2}{2n}\right )^n\,dx. \] You need to fully justify the use of whatever limit theorem you choose.

Raw LaTeX

Use your limit theorems to prove that the following limit exists
and to compute its value:
\[
  \lim_{n\to\infty}\int_{-\sqrt{n}}^{\sqrt{n}}
  \left(1-\frac{x^2}{2n}\right)^n\,dx.
\]
You need to fully justify the use of whatever limit theorem you choose.

Topic Tags

Central Limit Theorems Uniform Integrability and Truncation Methods