This set of Discrete Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Discrete Probability – Power Series”.

1. The explicit formula for the geometric sequence 3, 15, 75, 375,… is _______

a) 2*6! * 3^{n-1}

b) 3 * 5^{n-1}

c) 3! * 8^{n-1}

d) 7 * 4^{n-1}

View Answer

Explanation: The initial term is 3 and each subsequent term is the product of its previous term, the common ratio is 5. Thus the formula generating this sequence is a

_{n}= 3 * 5

^{n-1}.

2. The third term of a geometric progression with common ratio equal to half the initial term is 81. Determine the 12^{th} term.

a) 3^{12}

b) 4^{15}

c) 6^{8}

d) 5^{9}

View Answer

Explanation: Let the initial term be a and the common ratio r. The 3

^{rd}term is ar

^{2}= 27 and the initial term is a=3r so 3r

^{3}= 81 ⇒ r=3 ⇒ a=3. The a

_{12}= a * r

^{11}= 3 * 3

^{11}= 3

^{12}.

3. Which of the following series is called the “formal power series”?

a) b_{0}+b_{1}x+b_{2}x^{2}+…+b_{n}x^{n}

b) b_{1}x+b_{2}x^{2}+…+b_{n}x^{n}

c) 1/2b_{0}+1/3b_{1}x+1/4b_{2}x^{2}+…+1/nb_{n}x^{n}

d) n^{2}(b_{0}+b_{1}x+b_{2}x^{2}+…+b_{n}x^{n})

View Answer

Explanation: A formal power series is also called a “formal series”, of a field F is an infinite sequence b

_{0}, b

_{1}, b

_{2}, … over F. It is a function from the set of nonnegative integers to F i.e., 0, 1, 2, 3, … → F. A formal power series can also be written as b

_{0}+b

_{1}x+b

_{2}x

^{2}+…+b

_{n}x

^{n}.

4. sec(x) has a trigonometric series that is given by _______

a) ∞∑_{n=0} ((-1)^{n}E_{2n} / (2n)!)*x^{2n}

b) ∞∑_{n=0} ((-1)^{n}E_{2n})

c) ((-1)^{n}B_{2n} / (2n)!)*x^{2n}

d) ∞∑_{n=0} ((2n)!)*x^{2n+1}

View Answer

Explanation: A trigonometric series is an example of a Maclaurin series. Here, sec(x) can be represented as ∞∑

_{n=0}((-1)

^{n}E

_{2n}/ (2n)!)*x

^{2n}.

5. Determine the interval and radius of convergence for the power series: ∞∑_{n=1}7^{n}/n(3x−1)^{n-1}.

a) (2x+1)/6

b) 7|3x−1|

c) 5|x+1|

d) 3!*|4x−9|

View Answer

Explanation: Okay, let’s start off with the Ratio Test to get our hands on L = lim

_{n→ ∞}∣7

^{n+1}(3x−1)

^{n}/(n+1)

^{n}7

^{n}(3x−1)

^{n-1}∣=lim

_{n→∞}∣7

^{n}(3x−1)

^{n+1}∣=|3x−1|lim

_{n→∞}7

^{n}/(3n-1)=7|3x−1|.

6. Determine a power series representation for the function g(x)=ln(7−x).

a) ∞∑_{n}=0 x^{n+1}/7^{n+1}

b) ln(14)∞∑_{n}=0 x^{n+1}/7n

c) ln(7)∞∑_{n}=0 x^{n+1}/7^{n+1}

d) ln∞∑_{n}=0 x/7^{n+1}

View Answer

Explanation: We know that ∫1/7−x dx=−ln(7−x) and there is a power series representation for 1/7−x. So, ln(7−x)=−∫1/7−xdx

=−∫ ∞∑

_{n=0}x

^{n}/7

^{n+1}dx=C

⇒ ∞∑

_{n=0}x

^{n+1}/7

^{n+1}

So, the answer is, ln(7−x)=ln(7)∞∑

_{n=0}x

^{n+1}/7

^{n+1}.

7. An example of Maclaurin series is _______

a) ∞∑_{n=0} (x^{n}/n!)

b) ∞∑_{n=0} (x/5+n!)

c) ∞∑_{n=0} (x^{n+1}/(n-1)!)

d) (x^{n}/n)

View Answer

Explanation: The exponential function e

^{x}can described as ∞∑

_{n=0}(x

^{n}/n!) which is an example of a Maclaurin series. This series converges for all x.

8. Find the power series representation for the function f(x)=x/4−x.

a) ∞∑_{n=0}x^{n+1}/4^{n+1}

b) ∞∑_{n=0}x^{n+1}4^{n}

c) ∞∑_{n=0}x^{n}4^{n}

d) ∞∑_{n=0}x^{n+1}

View Answer

Explanation: So, again, we’ve got an x in the numerator. f(x)=x*1/4−x. If there is a power series representation for g(x)=1/4−x, there will be a power series representation for f(x). Suppose, g(x)=1/4*1/1−x

^{4}. To get a power series representation is to replace the x with x

^{4}. Doing this gives, g(x)=1/4 ∞∑n=0 x

^{n}/4

^{n}(x

^{n}/4 nprovided ∣x/4∣<1) ⇒ g(x) = 1/4 ∞∑

_{n=0}x

^{n}/4

^{n}= ∞∑

_{n=0}x

^{n}/4

^{n+1}. The interval of convergence for this series is, ∣x/4∣<1⇒1/4 |x|<1⇒|x|<4. Now, multiply g(x) by x and we have f(x)=x*1/4−x=x ⇒ ∞∑

_{n=0}x

^{n}/4

^{n+1}= ∞∑

_{n=0}x

^{n+1}/4

^{n+1}and the interval of convergence will be |x|<4.

9. What is the radius of convergence and interval of convergence for the power series ∞∑_{n=0}m!(2x-1)^{m}?

a) 3, 12

b) 1, 0.87

c) 2, 5.4

d) 0, 1/2

View Answer

Explanation: Suppose, L=lim

_{n→∞}|(m+1)!(2x+1)

^{m+1}/m!(2x+1)

^{m}|

= lim

_{m→∞}∣(m+1)m!(2x-1)/m!|

= |2x-1|lim

_{m→∞}(m+1)

So, this power series will only converge if x=1/2. We know that every power series will converge for x=a and in this case a=1/2. Remember that we get a from (x−a)

^{n}. In this case, the radius of convergence is R=0 and the interval of convergence is x=1/2.

10. Determine the radius of convergence and interval of convergence for the power series: ∞∑_{n=0} (x−7)^{n+1}/n^{n}.

a) 0, −1<x<1

b) ∞, −∞<x<∞

c) 1, −2<x<2

d) 2, −1<x<1

View Answer

Explanation: So, L=lim

_{n→∞}∣(x−7)

^{n+1}/n

^{n}∣

L=lim

_{n→∞}∣x−7/n∣

L=|x−7|lim

_{n→∞}1/n=0

So, since L=0<1 any of the value of x, this power series will converge for every x. In these cases, the radius of convergence is R=∞ and interval of convergence is −∞<x<∞.

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