AP_Calculus_BC_CH11_Exam

Question 1 of 16 | Series · Level 2

Which of the following sequences converges?

\(a_n = (-1)^n\)

\(a_n = \dfrac{(-1)^n}{n}\)

\(a_n = \sin(n)\)

\(a_n = n \sin(n)\)

Question 2 of 16 | Series · Level 2

Find the sum of the series \(\displaystyle\sum_{n=1}^{\infty} \dfrac{3}{4^n}\).

\(\dfrac{3}{4}\)

\(1\)

\(\dfrac{4}{3}\)

\(3\)

Question 3 of 16 | Series · Level 2

The series \(\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^p}\) converges if and only if:

\(p > 0\)

\(p \geq 1\)

\(p > 1\)

\(p \geq 2\)

Question 4 of 16 | Series · Level 3

Using the Integral Test, determine the convergence of \(\displaystyle\sum_{n=2}^{\infty} \dfrac{1}{n \ln n}\).

Converges

Diverges

Inconclusive

Cannot apply

Question 5 of 16 | Series · Level 3

Using the Comparison Test, \(\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^2 + 1}\):

Converges by comparison with \(\sum \dfrac{1}{n^2}\)

Diverges by comparison with \(\sum \dfrac{1}{n}\)

Converges by comparison with \(\sum \dfrac{1}{n}\)

Cannot be determined by comparison

Question 6 of 16 | Series · Level 3

The alternating series \(\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1}}{n^2}\) is:

Absolutely convergent

Conditionally convergent

Divergent

Cannot be determined

Question 7 of 16 | Series · Level 3

Use the Ratio Test on \(\displaystyle\sum_{n=1}^{\infty} \dfrac{n^2}{2^n}\). The series:

Converges, \(L = \dfrac{1}{2}\)

Diverges, \(L = 2\)

Inconclusive, \(L = 1\)

Converges, \(L = 0\)

Question 8 of 16 | Series · Level 3

The radius of convergence for \(\displaystyle\sum_{n=0}^{\infty} \dfrac{x^n}{n!}\) is:

\(R = 0\)

\(R = 1\)

\(R = e\)

\(R = \infty\)

Question 9 of 16 | Series · Level 4

Find the interval of convergence for \(\displaystyle\sum_{n=1}^{\infty} \dfrac{x^n}{n}\).

\((-1, 1)\)

\([-1, 1)\)

\((-1, 1]\)

\([-1, 1]\)

Question 10 of 16 | Series · Level 2

The Maclaurin series for \(\dfrac{1}{1-x}\) is:

\(\displaystyle\sum_{n=0}^{\infty} x^n\)

\(\displaystyle\sum_{n=0}^{\infty} (-1)^n x^n\)

\(\displaystyle\sum_{n=1}^{\infty} x^n\)

\(\displaystyle\sum_{n=0}^{\infty} n x^n\)

Question 11 of 16 | Series · Level 2

The Maclaurin series for \(e^x\) is:

\(\displaystyle\sum_{n=0}^{\infty} x^n\)

\(\displaystyle\sum_{n=0}^{\infty} \dfrac{x^n}{n!}\)

\(\displaystyle\sum_{n=1}^{\infty} \dfrac{x^n}{n}\)

\(\displaystyle\sum_{n=0}^{\infty} \dfrac{x^n}{n}\)

Question 12 of 16 | Series · Level 3

The coefficient of \(x^3\) in the Taylor series for \(\sin x\) centered at \(0\) is:

\(\dfrac{1}{3}\)

\(-\dfrac{1}{3}\)

\(\dfrac{1}{6}\)

\(-\dfrac{1}{6}\)

Question 13 of 16 | Series · Level 3

Using \(T_2(x)\) for \(\cos x\) at \(a = 0\), approximate \(\cos(0.1)\):

\(0.990\)

\(0.995\)

\(1.000\)

\(0.985\)

Question 14 of 16 | Series · Level 2

Which test is BEST for \(\displaystyle\sum_{n=1}^{\infty} \dfrac{3^n}{n!}\)?

Integral Test

Comparison Test

Ratio Test

Root Test

Question 15 of 16 | Series · Level 4

Consider the series \(\displaystyle\sum_{n=1}^{\infty} \dfrac{n}{3^n}\).

(a) Use the Ratio Test to determine whether the series converges or diverges. Show all steps.

Answer for 15(a)

(b) If the series converges, explain why absolute convergence and conditional convergence are the same in this case.

Answer for 15(b)

(c) Using the geometric series formula \(\displaystyle\sum_{n=0}^{\infty} x^n = \dfrac{1}{1-x}\) for \(|x| < 1\), differentiate both sides and find a closed form for \(\displaystyle\sum_{n=1}^{\infty} n x^{n-1}\).

Answer for 15(c)

Question 16 of 16 | Series · Level 4

Let \(f(x) = \ln(1 + x)\).

(a) Find the Maclaurin series for \(f(x) = \ln(1 + x)\) by integrating the series for \(\dfrac{1}{1+x}\).

Answer for 16(a)

(b) Determine the interval of convergence for the series found in part (a). Be sure to check the endpoints.

Answer for 16(b)

(c) Use the first four nonzero terms of the series to approximate \(\ln(1.5)\). Then use the Alternating Series Estimation Theorem to find an upper bound for the error in your approximation.

Answer for 16(c)

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