Science:Math Exam Resources/Courses/MATH101/April 2017/Question 03 (b)
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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 . 
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Hint 

Recall that when . 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. 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 Answer: 