题目
题目

MTH1030 -1035 - S1 2025 MTH1030/35 Week 10 lesson quiz: Power series defining functions

单项选择题

Which of the following statements is NOT true?

选项
A.a. The Maclaurin series of the sine function has an infinite radius of convergence.
B.b. For a positive radius of convergence there are four essentially different types of intervals of convergence.
C.c. The Maclaurin series of the exponential function has an infinite radius of convergence.
D.d. The radius of convergence of a power series is either 0, positive or infinite.
E.e. A power series either converges for no or infinitely many values.
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思路分析
Question restatement: Which of the following statements is NOT true? Option a: 'The Maclaurin series of the sine function has an infinite radius of convergence.' This is true because sin(x) has a power series with coefficients that yield a radius of convergence R = ∞, so it converges for all x. Option b: 'For a positive radius of convergence there are four essentially different types of intervals of convergence.' With a positive radius R > 0, the interval of convergence is an in......Login to view full explanation

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Which of the following statements is not correct?

In this question we work through the steps for solving the following problem: Find the interval of convergence and the radius of convergence of the power series ∞ ∑ n=0 (x−4)n 3n2    Solution outline: To find interval and radius of convergence, we usually start by using the Ratio Test: a) Evaluate the limit lim n→∞| an+1 an |=lim n→∞| (x−4)n+1 3(n+1)2 (x−4)n 3n2 |      |x-4| b) What does the Ratio Test and your answer in (a) tell you about convergence when |x−4|<1  ?    [ Select ] Nothing (because the ratio test inconclusive) The power series converges The power series diverges c) What does the Ratio Test and your answer in (a) tell you about convergence when |x−4|>1  ?    [ Select ] The power series converges Nothing (because the ratio test inconclusive) The power series diverges d) What does the Ratio Test and your answer in (a) tell you about convergence when |x−4|=1 ?    [ Select ] The power series diverges The power series converges Nothing (because the ratio test inconclusive) e) For what values of x is |x−4|<1 ? (Solve the inequality)  [ Select ] x < 5 -4 < x < 5 -4 < x < 4 3 < x < 5 2 < x < 6 f) When x=5  the power series in the question becomes ∞ ∑ n=0 1 3n2  . Does this series converge or diverge?    [ Select ] Diverges Converges g) When x=3   the power series in the question becomes ∞ ∑ n=0 (−1)n 3n2   . Does this series converge or diverge?    [ Select ] Diverges Converges h) Summarizing the results we have found, for what values of x does the power series in the question converge? [ Select ] Converges for x in the interval (-4,4) Converges for x in the interval [3,5] Converges for x in the interval (-4,5] Converges for x in the interval [3,5) Converges for x in the interval (3,5] Converges for x in the interval [-4,4] Note: This is the interval of convergence.

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