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AP®MathsCollege BoardCalculus BCExam QuestionsUnit 10: Infinite Sequences and SeriesTests for Divergence & Convergence

Tests for Divergence & Convergence (College Board AP® Calculus BC): Exam Questions

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1
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The following sequences all diverge. Which one cannot be shown to diverge by using the nth term test?

  • sum from n equals 1 to infinity of open parentheses negative 1 close parentheses to the power of n times n

  • sum from n equals 1 to infinity of fraction numerator n over denominator square root of n plus 1 end root end fraction

  • sum from n equals 1 to infinity of fraction numerator negative 1 over denominator n end fraction

  • sum from n equals 1 to infinity of fraction numerator n squared plus 2 n minus 7 over denominator 16 minus 9 n squared end fraction

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2
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For which of the following infinite series does the ratio test yield an inconclusive result?

  • sum from n equals 1 to infinity of 2 to the power of n over n

  • sum from n equals 1 to infinity of n over 2 to the power of n

  • sum from n equals 1 to infinity of fraction numerator n factorial over denominator 2 to the power of n end fraction

  • sum from n equals 1 to infinity of fraction numerator 2 n over denominator 2 n plus 3 end fraction

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3
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Which of the following infinite series converges?

  • sum from n equals 1 to infinity of 1 over n

  • sum from n equals 1 to infinity of fraction numerator n factorial over denominator e to the power of n end fraction

  • sum from n equals 1 to infinity of fraction numerator e to the power of n over denominator n factorial end fraction

  • sum from n equals 1 to infinity of fraction numerator n plus 1 over denominator 1 minus n end fraction

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4
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For any real number k it can be shown that

integral subscript 1 superscript infinity 1 over x to the power of k d x equals open curly brackets table row cell plus infinity comma space space space k less or equal than 1 end cell row cell fraction numerator 1 over denominator k minus 1 end fraction comma space space space k greater than 1 end cell end table close

Which of the following series can be shown to converge using only that result and the integral test?

  • sum from n equals 1 to infinity of cube root of n

  • sum from n equals 1 to infinity of fraction numerator 1 over denominator square root of n end fraction

  • sum from n equals 1 to infinity of fraction numerator 1 over denominator n square root of n end fraction

  • sum from n equals 1 to infinity of fraction numerator 1 over denominator n squared plus 1 end fraction

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5
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The alternating harmonic series sum from n equals 1 to infinity of open parentheses negative 1 close parentheses to the power of n plus 1 end exponent over n converges to ln 2. What is the minimum number of terms of the series that you would need to use to find an approximation for ln 2 that the alternating series error bound guarantees is not more than 0.001 away from the true value?

  • 99

  • 100

  • 999

  • 1000

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1
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Which of the following series converge?

I. space sum from n equals 1 to infinity of fraction numerator open parentheses n plus 3 close parentheses squared over denominator n open parentheses n plus 1 close parentheses open parentheses n plus 4 close parentheses end fraction

II. space sum from n equals 1 to infinity of open parentheses fraction numerator 3 to the power of n over denominator n factorial end fraction close parentheses

III. space sum from n equals 1 to infinity of fraction numerator open parentheses n plus 1 close parentheses factorial over denominator n to the power of 50 end fraction

  • II only

  • III only

  • I and II only

  • I, II, and III

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2
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Let sum from n equals 1 to infinity of a subscript n be a convergent series, with a subscript n greater than 0 for all n. Which of the following must be true?

  • sum from n equals 1 to infinity of n cubed a subscript n diverges

  • sum from n equals 1 to infinity of a subscript n over n converges

  • limit as n rightwards arrow infinity of open vertical bar a subscript n plus 1 end subscript over a subscript n close vertical bar equals 0

  • 0 less than open vertical bar a subscript n close vertical bar less than 1 for all n

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3
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Which of the following series diverge?

I. space sum from n equals 3 to infinity of open parentheses negative 1 close parentheses to the power of n plus 1 end exponent over n

II. space sum from n equals 1 to infinity of open parentheses fraction numerator square root of n plus 1 end root over denominator square root of n plus 1 end fraction close parentheses

III. space sum from n equals 2 to infinity of fraction numerator 3 over denominator n squared plus 2 end fraction

  • None

  • I only

  • II only

  • II and III only

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4
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For a particular value of q, limit as c rightwards arrow infinity of integral subscript 1 superscript c 1 over x to the power of q d x diverges to plus infinity. Which of the following must also be true?

  • sum from n equals 1 to infinity of 1 over n to the power of q converges

  • sum from n equals 1 to infinity of 1 over n to the power of q diverges

  • sum from n equals 1 to infinity of 1 over n to the power of q plus 1 end exponent converges

  • sum from n equals 1 to infinity of 1 over n to the power of q plus 1 end exponent diverges

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5
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The infinite series sum from n equals 1 to infinity of fraction numerator open parentheses negative 1 close parentheses to the power of n plus 1 end exponent over denominator n squared minus n plus 20 end fraction converges to a finite sum S. What is the minimum number of terms of the series that you would need to use to find an approximation for S that the alternating series error bound guarantees is not more than 1 over 400 away from the true value?

  • 18

  • 19

  • 20

  • 21

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1
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Which of the following statements about the series sum from n equals 1 to infinity of fraction numerator cos open parentheses n pi close parentheses over denominator square root of 2 n plus 7 end root end fraction is true?

  • The series converges absolutely.

  • The series converges conditionally.

  • The series converges but neither conditionally nor absolutely.

  • The series diverges.

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2
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Which of the following series converge?

I. space sum from n equals 1 to infinity of fraction numerator 1 over denominator 2 n end fraction open parentheses 5 over 3 close parentheses to the power of n

II. space sum from n equals 1 to infinity of open parentheses negative 1 close parentheses to the power of n plus 1 end exponent fraction numerator 1 over denominator 3 n plus 4 end fraction

III. space sum from n equals 2 to infinity of fraction numerator 1 over denominator n ln n end fraction

  • II only

  • I and II only

  • II and III only

  • I, II, and III

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3
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Which of the following series converge absolutely?

I. space sum from n equals 5 to infinity of fraction numerator open parentheses negative 1 close parentheses to the power of n over denominator square root of n minus 2 end fraction

II. space sum from n equals 1 to infinity of fraction numerator cos open parentheses 2 n pi close parentheses over denominator 2 pi open parentheses n factorial close parentheses end fraction

III. space 1 plus 1 minus 2 plus 1 half plus 1 half minus 1 plus 1 third plus 1 third minus 2 over 3 plus 1 fourth plus 1 fourth minus 1 half plus...

  • I only

  • II only

  • II and III only

  • I, II and III

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4
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Graph of a continuous, positive, decreasing function, curving down from upper left to bottom right. The graph is entirely in the first quadrant, starting from x=1 and continuing to a little bit beyond x=5.

Let f be a continuous function that is decreasing and positive on x greater or equal than 1. A part of the graph of f is shown in the diagram above. For all n greater or equal than 1, the series sum from n equals 1 to infinity of a subscript n has its nth term defined by a subscript n equals f open parentheses n close parentheses. If limit as c rightwards arrow infinity of integral subscript 1 superscript c f open parentheses x close parentheses d x equals 3, which of the following could be true?

  • sum from n equals 1 to infinity of a subscript n equals 2

  • sum from n equals 1 to infinity of a subscript n equals 3

  • sum from n equals 1 to infinity of a subscript n equals 4

  • sum from n equals 1 to infinity of a subscript n diverges

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5
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A function f is defined in power series form as f open parentheses x close parentheses equals sum from n equals 1 to infinity of fraction numerator x to the power of n over denominator n squared plus n plus 298 end fraction, which converges for open vertical bar x close vertical bar less or equal than 1. Let s subscript n open parentheses x close parentheses be the nth partial sum of the power series. What is the smallest number n for which the alternating series error bound guarantees that open vertical bar f open parentheses negative 1 close parentheses minus s subscript n open parentheses negative 1 close parentheses close vertical bar less or equal than 0.0001?

  • 97

  • 98

  • 99

  • 100

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