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

MATH101 April 2017

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Question 03 (b)
The interval of convergence of the power series $\sum _{n=0}^{\infty }nx^{n+1}$ is $(-1,1)$. (You don’t have to show this.) Find a formula (not involving infinite series) for $\sum _{n=0}^{\infty }nx^{n+1}$, valid for all $-1<x<1$.

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!

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Hint
In fact, we can use a function to express a infinite series in the interval of convergence. The best way is to connect it to geometric series.

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. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work.</span> If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. From the question we have $$\sum _{n=0}^{\infty }nx^{n+1}=x^{2}\sum _{n=0}^{\infty }nx^{n-1}.$$ For simplicity we denote ${\textstyle f(x)=\sum _{n=0}^{\infty }nx^{n-1}}$, if we know what ${\textstyle f(x)}$ exactly is, then we are done. First, we integrate at both sides it gives $$\int _{0}^{x}f(t)\ dt=\int _{0}^{x}\sum _{n=0}^{\infty }nt^{n-1}\ dt=\sum _{n=0}^{\infty }\int _{0}^{x}nt^{n-1}\ dt=\sum _{n=0}^{\infty }x^{n}={\frac {1}{1-x}}.$$ Where the last equality is from geometric series. Now differentiating both sides it gives $$f(x)={\frac {1}{(1-x)^{2}}}.$$ Finally, we get ${\textstyle \color {blue}\sum _{n=0}^{\infty }nx^{n+1}={\frac {x^{2}}{(1-x)^{2}}}.}$

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