# Real Analysis : Radius and Interval of Convergence - Power Series (3 Problems)

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Determine the radius of convergence and the exact interval of convergence of the following power series...

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The Radius and Interval of Convergence of Power Series are found. The solution is detailed and well presented. The response received a rating of "5" from the student who posted the question.

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Solution.

(a) For

We know that if a power series converges, then it converges absolutely. So, we consider . We have the ratio of the n+1 term and the n-th term is

If , then converges. So,

converges. So, we know that the radius of the convergence is .

Now we consider ...

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- BSc , Wuhan Univ. China
- MA, Shandong Univ.

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- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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- "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
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