Science:Math Exam Resources/Courses/MATH105/April 2012/Question 01 (c)
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Question 01 (c) 

Express the Taylor series of the function about x = 0 in summation notation. 
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 

First, use the fact from part (b) that provided r < 1, to get the power series of per integration. Then make a proper substitution for r. 
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 

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Please rate my easiness! It's quick and helps everyone guide their studies. For r<1 we use that and integrate both sides to obtain Since r < 1 we have 1r = 1r, therefore To solve for the integration constant C we plug in r = 0: so C = 0. So we obtain the intermediate result that To arrive at the summation notation, we noted that every term has the same sign (so there are no terms like floating around), every integer power of is included starting from , and each is divided by k. As a last step, the Taylor series for can be found by replacing by in the series above. We find:
