Science:Math Exam Resources/Courses/MATH101/April 2006/Question 07
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Question 07 

FullSolution Problems. Justify your answers and show all your work. Simplification of answers is not required. Evaluate . 
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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

The notation should be awfully reminiscent of Riemann sums. However, the way the sum is presented makes it look a bit mysterious from the Riemann sums we are used to. What or where is the value? What or where is the value? 
Hint 2 

Try factoring out an n from each of the two factors in the denominator. 
Hint 3 

The values should look like
and so with the above, we can deduce that

Hint 4 

Convert to an integral and now use partial fractions to take you home. 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

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. We proceed as in the hint (specifically hint 3). Notice that
From this, it is clear that and that we should probably set
and so and thus, combining this with the information, we have
and solving for gives . Hence, we have
We've reduced the problem to solving an integral. This integral is ripe for the partial fraction algorithm so let's apply that. Let
and this gives rise to
Plugging in in the above yields
and so . Plugging in into the equation gives
and so . Thus, we have
completing the question. 