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Question 07 (c) 

Find the Taylor series for the function . Write your answer in summation notation. (Hint: the Taylor series for may be helpful.) (Taylor 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 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 Taylor series for is . 
Hint 2 

The derivative of is . 
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 1 

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Please rate my easiness! It's quick and helps everyone guide their studies. Using the fact that is the derivative of , we can use the Taylor series of to obtain the Taylor series of . Specifically, . Therefore, to find we multiply the taylor series for by the factor . That is . 
Solution 2 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. The first solution assumes that the Taylor series for is not known in advance. The Taylor series for is known since it was given in the statement of a previous question. If the Taylor series for is known, then we can start the solution from the starting point . At this point, we multiply the sum by to obtain . 