MATH101 April 2017
• Q1 (a) • Q1 (b) • 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 • Q9 • Q10 • Q11 (a) • Q11 (b) • Q11 (c) •
Question 03 (b)
The interval of convergence of the power series is . (You don’t have to show this.) Find a formula (not involving infinite series) for , valid for all .
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!
Recall that when .
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From the question we have
Therefore, it is enough to find an explicit expression of .
For simplicity we denote .
Observe that . Using this, can be written as
Since the interval of convergence of the power series is we can reverse the order of summation and derivative on this interval to get
By the hint (which can be easily obtained from the explicit expression of a geometric series), this implies that
Therefore, computing the derivative based on the chain rule and power rule, the explicit formula for is given by
Combining with the first equation, we get