Infinite Sequences and Series MCQs in Calculus

By: Prof. Dr. Fazal Rehman Shamil | Last updated: February 12, 2025

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:

\[
\text{The Alternating Series Test is used to determine the convergence of an alternating series.}
\]
\[
\text{For an alternating series } \sum (-1)^n a_n, \text{ the test states that if } a_n \text{ is decreasing and } \lim_{n \to \infty} a_n = 0,
\]
\[
\text{then the series converges.}
\]
\[
\boxed{\text{Alternating Series Test}}
\]

More MCQs on Calculus

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  2. Limits and Derivatives MCQs in Calculus
  3. Differentiation Rules MCQs in Calculus
  4. Applications of Differentiation MCQs in Calculus
  5. Integrals MCQs in Calculus
  6. Applications of Integration MCQs in Calculus
  7. Techniques of Integration MCQs in Calculus
  8. Differential Equations MCQs in Calculus
  9. Parametric Equations and Polar Coordinates MCQs in Calculus
  10. Infinite Sequences and Series MCQs in Calculus
  11. Vectors and the Geometry of Space MCQs in Calculus
  12. Vector Functions MCQs in Calculus
  13. Partial Derivatives MCQs in Calculus
  14. Multiple Integrals MCQs in Calculus
  15. Vector Calculus MCQs in Calculus
  16. Second-Order Differential Equations MCQs in Calculus

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