Science:Math Exam Resources/Courses/MATH105/April 2011/Question 09 (c)

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MATH105 April 2011

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Question 09 (c)
Each of the short-answer questions below is worth 5 points. Put your answer in the box provided and show your work. No credit will be given for the answer without the correct accompanying work. Express the limit $\lim _{n\rightarrow \infty }{\frac {1}{n}}\sum _{k=1}^{n}\ln \left(1+{\frac {k}{n}}\right)$ as a definite integral, and leave it in the form of an integral. Do not evaluate it!

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
The Riemann sum formula using the right end points for a function $f(x)$ is $f(x)=R_{n}=\sum _{k=1}^{n}f(x_{k})\Delta x$ where $x_{k}=a+k\Delta x\quad \quad \Delta x={\frac {b-a}{n}}$

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution 1

The Riemann sum formula using the right end points for a function $f(x)$ is $f(x)=R_{n}=\sum _{k=1}^{n}f(x_{k})\Delta x$ where $x_{k}=a+k\Delta x\quad \quad \Delta x={\frac {b-a}{n}}$ In our example, comparing the terms, we see that $\Delta x={\frac {1}{n}}$ and so $b-a=1$ Also, we have that $f(x_{k})=\ln \left(1+{\frac {k}{n}}\right)$ So taking $f(x)=\ln(1+x)$, we have $x_{k}=a+k\Delta x=a+{\frac {k}{n}}={\frac {k}{n}}$ so we have $a=0$ and hence since $b-a=1$ and so $b=b-0=1$. Plugging all this information in, we have $\lim _{n\rightarrow \infty }{\frac {1}{n}}\sum _{k=1}^{n}\ln \left(1+{\frac {k}{n}}\right)=\int _{0}^{1}\ln(1+x)\,dx$

Solution 2

The Riemann sum formula using the right end points for a function $f(x)$ is $f(x)=R_{n}=\sum _{k=1}^{n}f(x_{k})\Delta x$ where $x_{k}=a+k\Delta x\quad \quad \Delta x={\frac {b-a}{n}}$ In our example, comparing the terms, we see that $\Delta x={\frac {1}{n}}$ and so $b-a=1$ Also, we have that $f(x_{k})=\ln \left(1+{\frac {k}{n}}\right)$ So taking $f(x)=\ln(x)$, we have $x_{k}=a+k\Delta x=a+{\frac {k}{n}}=1+{\frac {k}{n}}$ so we have $a=1$ and hence since $b-a=1$ and so $b-1=1$ giving $b=2$. Plugging all this information in, we have $\lim _{n\rightarrow \infty }{\frac {1}{n}}\sum _{k=1}^{n}\ln \left(1+{\frac {k}{n}}\right)=\int _{1}^{2}\ln(x)\,dx$

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