Science:Math Exam Resources/Courses/MATH101/April 2008/Question 01 (c)

MATH101 April 2008

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Question 01 (c)
Express $\lim _{n\to \infty }{\frac {1}{n}}\sum _{i=1}^{n}{\frac {1}{1+({\tfrac {i}{n}})^{2}}}$ as a definite integral. Do not evaluate the integral.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
This is a Riemann sum. Recall that $\displaystyle \Delta x={\frac {b-a}{n}}$ and that $\displaystyle x_{i}=a+i\Delta x.$ Then do a pattern match.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. From the question, we immediately see that $\displaystyle \Delta x={\frac {1}{n}}$ Now, to determine $\displaystyle x_{i}$, notice that we have that the function is of the form $\displaystyle f(x_{i})={\frac {1}{1+({\tfrac {i}{n}})^{2}}}.$ The only composition of functions is underneath the square sign. So we must have that $\displaystyle f(x)={\frac {1}{1+x^{2}}}$ and hence that $\displaystyle x_{i}={\frac {i}{n}}$ As $\displaystyle x_{i}=a+i\Delta x$, we have that $\displaystyle a=0$ and hence since $\displaystyle \Delta x={\frac {1}{n}}={\frac {b-a}{n}}={\frac {b}{n}}$ we have that $\displaystyle b=1$. Combining the above gives $\lim _{n\to \infty }{\frac {1}{n}}\sum _{i=1}^{n}{\frac {1}{1+({\tfrac {i}{n}})^{2}}}=\int _{0}^{1}{\frac {1}{1+x^{2}}}\,dx$ which completes the question.

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