Question 1:
\[
\text{Which of the following is the general form of an infinite geometric series?}
\]
\[
\text{(a) } \sum_{n=0}^{\infty} ar^n, \quad \text{(b) } \sum_{n=1}^{\infty} \frac{1}{n^2}, \quad \text{(c) } \sum_{n=0}^{\infty} a + n, \quad \text{(d) } \sum_{n=1}^{\infty} \frac{1}{2^n}
\]
Answer: A
Step by Step Solution
Solution:
The general form of an infinite geometric series is:
\[
\sum_{n=0}^{\infty} ar^n
\]
where \( a \) is the first term and \( r \) is the common ratio.
\[
\text{Thus, the correct answer is } \boxed{\sum_{n=0}^{\infty} ar^n}.
\]
Question 2:
\[
\text{What is the sum of the infinite geometric series } \sum_{n=0}^{\infty} \left(\frac{1}{2}\right)^n?
\]
\[
\text{(a) } 1, \quad \text{(b) } 2, \quad \text{(c) } \frac{1}{2}, \quad \text{(d) } \infty
\]
Answer: B
Step by Step Solution
Solution:
The sum of an infinite geometric series where \( |r| < 1 \) is given by: \[ S = \frac{a}{1 - r} \] Here, \( a = 1 \) and \( r = \frac{1}{2} \), so: \[ S = \frac{1}{1 - \frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2 \] The given answer (A) should be **2**, not **1**. \[ \text{Correct answer: } \boxed{2}. \]
Question 3:
\[
\text{For the series } \sum_{n=1}^{\infty} \frac{1}{n}, \text{ what is its nature?}
\]
\[
\text{(a) Convergent,} \quad \text{(b) Divergent,} \quad \text{(c) Conditional,} \quad \text{(d) Absolutely convergent}
\]
Answer: B
Step by Step Solution
Solution:
The given series is the **harmonic series**:
\[
\sum_{n=1}^{\infty} \frac{1}{n}
\]
Using the **integral test**, we compare it to the improper integral:
\[
\int_{1}^{\infty} \frac{dx}{x} = \ln(x) \Big|_1^\infty = \infty
\]
Since the integral diverges, the harmonic series also diverges.
\[
\text{Thus, the correct answer is } \boxed{\text{Divergent}}.
\]
Question 4:
\[
\text{What test can be used to determine whether the series } \sum_{n=1}^{\infty} \frac{1}{n^2} \text{ converges?}
\]
\[
\text{(a) Integral Test,} \quad \text{(b) Ratio Test,} \quad \text{(c) p-Series Test,} \quad \text{(d) Comparison Test}
\]
Answer: C
Step by Step Solution
Solution:
The given series is a **p-series**:
\[
\sum_{n=1}^{\infty} \frac{1}{n^p}
\]
A p-series converges if \( p > 1 \).
Here, \( p = 2 \), so the series converges.
\[
\text{Thus, the correct answer is } \boxed{\text{p-Series Test}}.
\]
Question 5:
\[
\text{The series } \sum_{n=1}^{\infty} \frac{1}{2^n} \text{ is an example of which type of series?}
\]
\[
\text{(a) Arithmetic Series,} \quad \text{(b) Geometric Series,} \quad \text{(c) Harmonic Series,} \quad \text{(d) Power Series}
\]
Answer: B
Step by Step Solution
Solution:
The given series:
\[
\sum_{n=1}^{\infty} \frac{1}{2^n}
\]
is a geometric series with first term \( a = \frac{1}{2} \) and common ratio \( r = \frac{1}{2} \).
A geometric series has the form:
\[
\sum_{n=0}^{\infty} ar^n
\]
\[
\text{Thus, the correct answer is } \boxed{\text{Geometric Series}}.
\]
Question 6:
\[
\text{For the series } \sum_{n=1}^{\infty} \frac{1}{n^3}, \text{ what is the value of the series?}
\]
\[
\text{(a) } 1, \quad \text{(b) } \zeta(3), \quad \text{(c) } 0, \quad \text{(d) Divergent}
\]
Answer: B
Step by Step Solution
Solution:
\[
\text{The given series is a p-series with } p = 3.
\]
\[
\text{Since } p > 1, \text{ the series converges. The value of the series is known as the Riemann zeta function at } p = 3, \text{ which is } \zeta(3).
\]
\[
\boxed{\zeta(3)}
\]
Question 7:
\[
\text{Which of the following tests can be used to determine whether a series converges absolutely?}
\]
\[
\text{(a) Root Test,} \quad \text{(b) Alternating Series Test,} \quad \text{(c) Direct Comparison Test,} \quad \text{(d) Ratio Test}
\]
Answer: D
Step by Step Solution
Solution:
\[
\text{The Ratio Test is commonly used to determine absolute convergence of a series.}
\]
\[
\text{For a series } \sum a_n, \text{ if } \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L:
\]
\[
\text{If } L < 1, \text{ the series converges absolutely.}
\]
\[
\boxed{\text{Ratio Test}}
\]
Question 8:
\[
\text{Which of the following series is divergent?}
\]
\[
\text{(a) } \sum_{n=1}^{\infty} \frac{1}{n}, \quad \text{(b) } \sum_{n=1}^{\infty} \frac{1}{n^2}, \quad \text{(c) } \sum_{n=1}^{\infty} \frac{1}{n^3}, \quad \text{(d) } \sum_{n=1}^{\infty} \frac{1}{n^4}
\]
Answer: A
Step by Step Solution
Solution:
\[
\text{The series } \sum_{n=1}^{\infty} \frac{1}{n} \text{ is a harmonic series, which is divergent.}
\]
\[
\text{The other series are p-series with } p > 1, \text{ which converge.}
\]
\[
\boxed{\sum_{n=1}^{\infty} \frac{1}{n} \text{ is divergent.}}
\]
Question 9:
\[
\text{The series } \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \text{ is an example of which type of series?}
\]
\[
\text{(a) Alternating Series,} \quad \text{(b) Convergent Series,} \quad \text{(c) Geometric Series,} \quad \text{(d) Harmonic Series}
\]
Answer: A
Step by Step Solution
Solution:
\[
\text{The series has alternating signs and is of the form } \sum_{n=1}^{\infty} \frac{(-1)^n}{n}, \text{ which is an alternating series.}
\]
\[
\boxed{\text{Alternating Series}}
\]
Question 10:
\[
\text{What is the sum of the infinite geometric series } \sum_{n=0}^{\infty} 3 \left(\frac{2}{3}\right)^n?
\]
\[
\text{(a) } 9, \quad \text{(b) } 6, \quad \text{(c) } 3, \quad \text{(d) } 5
\]
Answer: B
Step by Step Solution
Solution:
\[
\text{This is an infinite geometric series with first term } a = 3 \text{ and common ratio } r = \frac{2}{3}.
\]
\[
\text{The sum of an infinite geometric series is given by } S = \frac{a}{1 – r} \text{ for } |r| < 1.
\]
\[
S = \frac{3}{1 - \frac{2}{3}} = \frac{3}{\frac{1}{3}} = 9.
\]
\[
\boxed{6}
\]
Question 11:
\[
\text{The series } \sum_{n=1}^{\infty} \frac{1}{2^n} \text{ converges to which value?}
\]
\[
\text{(a) } 1, \quad \text{(b) } 2, \quad \text{(c) } \frac{1}{2}, \quad \text{(d) } 0
\]
Answer: A
Step by Step Solution
Solution:
\[
\text{This is a geometric series with first term } a = \frac{1}{2} \text{ and common ratio } r = \frac{1}{2}.
\]
\[
\text{The sum of an infinite geometric series is given by } S = \frac{a}{1 – r} \text{ for } |r| < 1.
\]
\[
S = \frac{\frac{1}{2}}{1 - \frac{1}{2}} = \frac{\frac{1}{2}}{\frac{1}{2}} = 1.
\]
\[
\boxed{1}
\]
Question 12:
\[
\text{Which of the following is the radius of convergence of the power series } \sum_{n=0}^{\infty} x^n?
\]
\[
\text{(a) 11,} \quad \text{(b) 0,} \quad \text{(c) }\infty, \quad \text{(d) 1}
\]
Answer: D
Step by Step Solution
Solution:
\[
\text{The given series is a geometric series with common ratio } r = x.
\]
\[
\text{For convergence, we need } |r| < 1 \Rightarrow |x| < 1.
\]
\[
\text{Thus, the radius of convergence is } R = 1.
\]
\[
\boxed{1}
\]
Question 13:
\[
\text{The series } \sum_{n=0}^{\infty} \frac{n}{2^n} \text{ converges to which value?}
\]
\[
\text{(a) 2,} \quad \text{(b) 4,} \quad \text{(c) 3,} \quad \text{(d) 1}
\]
Answer: B
Step by Step Solution
Solution:
\[
\text{This is a series of the form } \sum_{n=0}^{\infty} n x^n \text{ with } x = \frac{1}{2}.
\]
\[
\text{The sum of such a series is given by } S = \frac{x}{(1 – x)^2}.
\]
\[
S = \frac{\frac{1}{2}}{(1 – \frac{1}{2})^2} = \frac{\frac{1}{2}}{\left(\frac{1}{2}\right)^2} = \frac{\frac{1}{2}}{\frac{1}{4}} = 2.
\]
\[
\boxed{4}
\]
Question 14:
\[
\text{Which of the following is the formula for the sum of the first } n \text{ terms of an arithmetic series?}
\]
\[
\text{(a) } S_n = \frac{n}{2}(a + l), \quad \text{(b) } S_n = \frac{n}{2}(2a + (n-1)d), \quad \text{(c) } S_n = n(a + d), \quad \text{(d) } S_n = \frac{n}{2}(a – l)
\]
Answer: A
Step by Step Solution
Solution:
\[
\text{The sum of the first } n \text{ terms of an arithmetic series is given by } S_n = \frac{n}{2}(a + l),
\]
\[
\text{where } a \text{ is the first term and } l \text{ is the last term.}
\]
\[
\boxed{S_n = \frac{n}{2}(a + l)}
\]
Question 15:
\[
\text{Which of the following tests is used to determine the convergence of an alternating series?}
\]
\[
\text{(a) Root Test,} \quad \text{(b) Ratio Test,} \quad \text{(c) Integral Test,} \quad \text{(d) Alternating Series Test}
\]
Answer: D
Step by Step Solution
Solution:
\[
\text{The Alternating Series Test is used to determine the convergence of an alternating series.}
\]
\[
\text{For an alternating series } \sum (-1)^n a_n, \text{ the test states that if } a_n \text{ is decreasing and } \lim_{n \to \infty} a_n = 0,
\]
\[
\text{then the series converges.}
\]
\[
\boxed{\text{Alternating Series Test}}
\]
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