Science:Math Exam Resources/Courses/MATH101/April 2015/Question 06 (a)
• Q1 (a) • Q1 (b) (i) • Q1 (b) (ii) • Q1 (b) (iii) • Q1 (c) • Q1 (d) • Q2 (a) • Q2 (b) • Q2 (c) (i) • Q2 (c) (ii) • Q2 (c) (iii) • Q2 (c) (iv) • Q3 (a) • Q3 (b) • Q4 (a) • Q4 (b) • Q5 (a) • Q5 (b) • Q6 (a) • Q6 (b) • Q7 (a) • Q7 (b) • Q8 (a) • Q8 (b) • Q9 (a) • Q9 (b) • Q10 (a) • Q10 (b) • Q11 (a) • Q11 (b) •
Question 06 (a) |
---|
Find the Maclaurin series for . |
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 |
---|
Find the Maclaurin series of , and try to use that to write the series for the integrand. |
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.
|
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. Let , we start by finding the Maclaurin series for . One way is by finding the 1st, 2nd, 3rd, ... derivatives of and at each step evaluate , and write the series. However, to avoid the cumbersome computation of the differentiation we use the fact that i.e. is the anti-derivative of , so . On the other hand, we know that . If we take , then we have
Therefore,
From this series we now build the series for the original function:
|