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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Tests for Divergence & Convergence

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1
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1 mark

Determine whether or not the series sum from x equals 1 to infinity of fraction numerator n over denominator 5 n plus 1 end fraction equals 1 over 6 plus 2 over 11 plus 3 over 16 plus 4 over 21 plus... converges. Justify your answer.

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2
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2 marks

Determine whether or not the series 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 4 over denominator 3 n squared plus 1 end fraction equals 1 minus 4 over 13 plus 1 over 7 minus 4 over 49 plus... converges. Justify your answer.

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3
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2 marks

Use the ratio test to determine whether or not the series sum from n equals 1 to infinity of fraction numerator e to the power of n over denominator n factorial end fraction converges.

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4
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2 marks

Use the limit comparison test to determine whether the series sum from n equals 1 to infinity of fraction numerator n plus 4 over denominator 3 n squared minus n minus 1 end fraction equals 5 plus 2 over 3 plus 7 over 23 plus 8 over 43 plus... converges or diverges.

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5
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2 marks

The 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 the value ln 2. Explain why this series is conditionally convergent rather than absolutely convergent.

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1
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3 marks

State the necessary conditions for using the integral test to determine whether or not the series sum from n equals 0 to infinity of 1 over 3 to the power of 2 n end exponent converges. Use the integral test to show that sum from n equals 0 to infinity of 1 over 3 to the power of 2 n end exponent converges.

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2
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1 mark

A function f is defined in power series form by f open parentheses x close parentheses equals sum from n equals 0 to infinity of open parentheses negative 1 close parentheses to the power of n 3 x to the power of 2 n plus 1 end exponent. Explain whether or not the series will converge for f open parentheses 1 close parentheses.

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3
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2 marks

Given that sum from n equals 0 to infinity of 1 over e to the power of n is a convergent series, use the limit comparison test to show that sum from n equals 0 to infinity of fraction numerator open parentheses negative 1 close parentheses to the power of n over denominator 4 e to the power of n minus 3 end fraction converges absolutely.

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4
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3 marks

Give a value of space p such that sum from n equals 1 to infinity of open parentheses negative 1 close parentheses to the power of n over n to the power of p converges, but sum from n equals 1 to infinity of 1 over n to the power of 3 p end exponent diverges. Give reasons why your value of space p is correct.

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5
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2 marks

Consider the convergent 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 n over denominator open parentheses 2 n minus 1 close parentheses factorial end fraction equals 1 minus fraction numerator 2 over denominator 3 factorial end fraction plus fraction numerator 3 over denominator 5 factorial end fraction minus fraction numerator 4 over denominator 7 factorial end fraction plus... plus fraction numerator open parentheses negative 1 close parentheses to the power of n plus 1 end exponent n over denominator open parentheses 2 n minus 1 close parentheses factorial end fraction plus.... Show that 83 over 120 approximates the value of the series sum with error less than 1 over 1000.

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6
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1 mark

The first two terms of the series g(1) = \sum_{n = 0}^{\infty} \frac{(- 1)^{n}}{2 e^{n} + 3} are used to approximate g(1). Use the alternating series error bound to determine an upper bound on the error of the approximation.

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7
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2 marks

Determine whether the series \sum_{n = 5}^{\infty} \frac{3}{2 n^{2} - 7 n + 5} converges or diverges. State the conditions of the test used for determining convergence or divergence.

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8
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2 marks

The Maclaurin series for a function f is given by \sum_{n = 1}^{\infty} \frac{(n + 1) x^{n}}{n^{2} 6^{n}} and converges to f(x) for all x in the interval of convergence. It can be shown that the Maclaurin series for f has a radius of convergence of 6.

Determine whether the Maclaurin series for f converges or diverges at x = 6. Give a reason for your answer.

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1
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2 marks

A function f is given in power series form as f open parentheses x close parentheses equals sum from n equals 1 to infinity of fraction numerator open parentheses n plus 4 close parentheses open parentheses x minus 2 close parentheses to the power of n over denominator n squared 3 to the power of n end fraction. Determine whether the series for f converges or diverges at x equals 5. Give a reason for your answer.

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2
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4 marks

Determine whether or not the 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 open parentheses 3 n minus 2 close parentheses over denominator 2 n squared plus 1 end fraction equals 1 third minus 4 over 9 plus 7 over 19 minus 10 over 33 plus... converges. Justify your answer.

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3
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4 marks

Use the integral test to prove that the p-series sum from n equals 1 to infinity of 1 over n to the power of p converges for p greater than 1 and diverges for 0 less than p less or equal than 1. Be sure to state the necessary conditions for using the integral test to determine convergence or divergence of these series.

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4
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3 marks

Determine whether the series sum from n equals 6 to infinity of fraction numerator 19 over denominator 6 n squared minus 25 n plus 11 end fraction converges or diverges. State and confirm the conditions of the test used for determining convergence or divergence.

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5
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5 marks

Determine whether the series sum from n equals 2 to infinity of fraction numerator open parentheses negative 1 close parentheses to the power of n ln n over denominator n end fraction converges absolutely, converges conditionally, or diverges. Justify your answer.

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6a
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3 marks

The function g has derivatives of all orders for all real numbers. The Maclaurin series for g is given by g(x) = \sum_{n = 0}^{\infty} \frac{(- 1)^{n} x^{n}}{2 e^{n} + 3} on its interval of convergence.

State the conditions necessary to use the integral test to determine convergence of the series \sum_{n = 0}^{\infty} \frac{1}{e^{n}}. Use the integral test to show that \sum_{n = 0}^{\infty} \frac{1}{e^{n}} converges.

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6b
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2 marks

Use the limit comparison test with the series \sum_{n = 0}^{\infty} \frac{1}{e^{n}} to show that the series g(1) = \sum_{n = 0}^{\infty} \frac{(- 1)^{n}}{2 e^{n} + 3} converges absolutely.

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