AP®MathsCollege BoardCalculus BCExam QuestionsUnit 10: Infinite Sequences and SeriesIntroduction to Infinite Series

Introduction to Infinite Series (College Board AP® Calculus BC): Exam Questions

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Introduction to Infinite Series

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1
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3 marks

A sequence open curly brackets a subscript n close curly brackets is defined by a subscript n equals fraction numerator 2 n squared plus 3 over denominator n end fraction for n greater or equal than 1. The sequence open curly brackets s subscript n close curly brackets is the sequence of partial sums for the associated series sum from n equals 1 to infinity of a subscript n.

Write down a subscript 7 and s subscript 3.

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2a
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1 mark

Show that sum from n equals 0 to infinity of 1 over e to the power of n is a geometric series.

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2b
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2 marks

Determine whether sum from n equals 0 to infinity of 1 over e to the power of n converges or diverges. If it converges, find the series sum.

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3
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2 marks

Determine the range of q values for which the series sum from n equals 1 to infinity of open parentheses fraction numerator 1 over denominator square root of n end fraction close parentheses to the power of q converges, and the range of q values for which it diverges.

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1
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3 marks

A sequence open curly brackets a subscript n close curly brackets is defined by a subscript n equals open parentheses n minus 1 close parentheses squared over 9900 for n greater or equal than 1. The sequence open curly brackets s subscript n close curly brackets is the sequence of partial sums for the associated series sum from n equals 1 to infinity of a subscript n.

Find the value of s subscript 101 minus s subscript 99.

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2
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2 marks

It can be shown that sum from n equals 1 to infinity of 1 over n squared equals pi squared over 6 and 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 equals ln 2.

Determine the value of 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 times 7 n squared minus 5 n over denominator n cubed end fraction.

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3
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3 marks

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

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4
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2 marks

Determine whether the infinite series sum from n equals 1 to infinity of 2 over 3 to the power of 2 n end exponent converges or diverges. If it converges, find the sum of the series.

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5
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3 marks

Consider the number 0.0 4 with dot on top 2 3 with dot on top equals 0.0423423423423.... By first writing the number in the form of a geometric series, find the value of 0.0 4 with dot on top 2 3 with dot on top as a fraction in lowest terms.

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1
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3 marks

Determine whether or not the infinite series sum from n equals 2 to infinity of fraction numerator 3 to the power of n minus 2 to the power of n over denominator 6 to the power of n end fraction converges, and if it converges determine its value.

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2
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1 mark

A function f is given in power series form as 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 end exponent. Find the value off open parentheses 1 third close parentheses.

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3
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4 marks

A sequence open curly brackets a subscript n close curly brackets is defined by a subscript n equals fraction numerator 4 over denominator n squared plus 2 n end fraction for n greater or equal than 1. The sequence open curly brackets s subscript n close curly brackets is the sequence of partial sums for the associated series sum from n equals 1 to infinity of a subscript n.

Use partial fractions to find an expression for s subscript n in terms of n. Explain why sum from n equals 1 to infinity of fraction numerator 4 over denominator n squared plus 2 n end fraction converges and find its sum.

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