Science:Math Exam Resources/Courses/MATH103/April 2009/Question 02
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Question 02 

Using the fact that for x near zero, find the Taylor series for the function valid for x near zero. 
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 

Note that If you multiply the two corresponding Taylor series, what will be the coefficient in front of the x^{k} term? Start with small values of k and find a pattern. 
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. We just plug in the functions
Next we start to multiply the two infinite sums term by term, and sort the result by powers of x. Failed to parse (unknown function "\begin{align}"): {\displaystyle \begin{align} \sum_{k=0}^\infty x^k\cdot \sum_{m=0}^\infty x^m &= {\color{red} \left(1+x+x^2+x^3+\dots\right)}{\color{blue} \left(1+x+x^2+x^3+\dots\right)} \\ &= ({\color{red} 1} \cdot {\color{blue} 1}) + ({\color{red} x} \cdot {\color{blue} 1} + {\color{red} 1} \cdot {\color{blue} x}) + ({\color{red} 1} \cdot {\color{blue} x^2} + {\color{red} x} \cdot {\color{blue} x} + {\color{red} x^2} \cdot {\color{blue} 1}) \\ &\quad + ({\color{red} 1} \cdot {\color{blue} x^3} + {\color{red} x} \cdot {\color{blue} x^2} + {\color{red} x^2} \cdot {\color{blue} x} + {\color{red} x^3} \cdot {\color{blue} 1} ) + \dots\\ &= 1+2x + 3x^2 + 4x^3 + \dots \end{align} } Now we can see the pattern: There are always k + 1 combinations that give the term x_{k}. Hence the Taylor series is given by . 