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Sequences & Series: 25 MCQs with answers

All 25 multiple-choice questions from this kit, free and with no sign-up.

Full kit · everything

Warm-up · 17 questions

Q1. What is the limit of the sequence $a_n = \frac{1}{n}$ as $n \to \infty$?

Beginner
A Does not exist B $\infty$ C 1 D 0
Show answer & explanation

0

As $n$ grows without bound, $\frac{1}{n}$ approaches 0.

Q2. Which of the following series is a geometric series?

Beginner
A $\sum_{n=1}^{\infty} \frac{1}{2^n}$ B $\sum_{n=1}^{\infty} \frac{1}{n!}$ C $\sum_{n=1}^{\infty} \frac{1}{n^2}$ D $\sum_{n=1}^{\infty} \frac{1}{n}$
Show answer & explanation

$\sum_{n=1}^{\infty} \frac{1}{2^n}$

A geometric series has the form $\sum ar^{n-1}$. Here $a=1/2$ and $r=1/2$.

Q3. The series $\sum_{n=1}^{\infty} \frac{1}{n}$ is known as the:

Beginner
A Geometric series B Harmonic series C Alternating series D Telescoping series
Show answer & explanation

Harmonic series

The harmonic series is $\sum_{n=1}^{\infty} \frac{1}{n}$, which diverges.

Q4. What is the $n$th term test for divergence?

Beginner
A If $\lim_{n\to\infty} a_n \neq 0$, then $\sum a_n$ diverges. B If $\lim_{n\to\infty} a_n = 0$, then $\sum a_n$ converges. C If $\sum a_n$ converges, then $\lim_{n\to\infty} a_n = 1$. D If $\lim_{n\to\infty} a_n = L$, then $\sum a_n = L$.
Show answer & explanation

If $\lim_{n\to\infty} a_n \neq 0$, then $\sum a_n$ diverges.

The $n$th term test states that if the limit of the terms is not zero, the series diverges.

Q5. Which of the following sequences is bounded?

Beginner
A $a_n = 2^n$ B $a_n = (-1)^n$ C $a_n = \ln n$ D $a_n = n$
Show answer & explanation

$a_n = (-1)^n$

The sequence $(-1)^n$ takes only values $-1$ and $1$, so it is bounded.

Q6. Every convergent sequence is bounded.

Beginner
A True B False
Show answer & explanation

True

A convergent sequence must be bounded; otherwise it could not approach a finite limit.

Q7. If a series converges, then its terms must approach zero.

Beginner
A True B False
Show answer & explanation

True

This is the $n$th term test: convergence implies $\lim_{n\to\infty} a_n = 0$.

Q8. What is the definition of a sequence?

Beginner
A A geometric figure B A collection of numbers arranged in a specific order C A single number D A sum of numbers
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A collection of numbers arranged in a specific order

A sequence is defined as a collection of numbers arranged in a specific order.

Q9. Which of the following represents a convergent sequence?

Beginner
A The sequence defined by $a_n = (-1)^n$ B The sequence defined by $a_n = n^2$ C The sequence defined by $a_n = \frac{1}{n}$ as $n \to \infty$ D The sequence defined by $a_n = n$ as $n \to \infty$
Show answer & explanation

The sequence defined by $a_n = \frac{1}{n}$ as $n \to \infty$

The sequence $a_n = \frac{1}{n}$ converges to 0 as $n \to \infty$.

Q10. What is the sum of the first $n$ terms of a geometric series with first term $a$ and common ratio $r$?

Beginner
A $S_n = a + r + r^2 + ... + r^{n-1}$ B $S_n = n \cdot a$ C $S_n = a \cdot \frac{1 - r^n}{1 - r}$ D $S_n = a + n \cdot r$
Show answer & explanation

$S_n = a \cdot \frac{1 - r^n}{1 - r}$

The sum of the first $n$ terms of a geometric series is given by $S_n = a \cdot \frac{1 - r^n}{1 - r}$.

Q11. Which of the following is a property of convergent series?

Beginner
A The series must be alternating B The series must have a finite sum C The terms must be increasing D The terms must approach a non-zero limit
Show answer & explanation

The series must have a finite sum

A convergent series is defined as one that has a finite sum.

Q12. What is the limit of the sequence $b_n = 2 - \frac{1}{n}$ as $n \to \infty$?

Beginner
A 1 B Infinity C 0 D 2
Show answer & explanation

2

As $n \to \infty$, $b_n$ approaches 2.

Q13. Which of the following series diverges?

Beginner
A $\sum_{n=1}^{\infty} \frac{1}{n^2}$ B $\sum_{n=1}^{\infty} \frac{1}{2^n}$ C $\sum_{n=1}^{\infty} \frac{1}{n}$ D $\sum_{n=1}^{\infty} \frac{1}{n^3}$
Show answer & explanation

$\sum_{n=1}^{\infty} \frac{1}{n}$

The series $\sum_{n=1}^{\infty} \frac{1}{n}$ diverges, known as the harmonic series.

Q14. What is the formula for the $n$th term of an arithmetic sequence with first term $a$ and common difference $d$?

Beginner
A $a_n = a + (n - 1)d$ B $a_n = a + d^n$ C $a_n = a \cdot d^n$ D $a_n = a + nd$
Show answer & explanation

$a_n = a + (n - 1)d$

The $n$th term of an arithmetic sequence is given by $a_n = a + (n - 1)d$.

Q15. What is the convergence criterion for the series $\sum_{n=1}^{\infty} \frac{1}{n^p}$?

Beginner
A It converges for $p > 1$ B It diverges for all $p$ C It converges for $p \leq 1$ D It converges for $p < 0$
Show answer & explanation

It converges for $p > 1$

The series $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges if and only if $p > 1$.

Q16. Which of the following is an example of a divergent sequence?

Beginner
A The sequence defined by $c_n = \frac{1}{n^2}$ B The sequence defined by $c_n = n$ C The sequence defined by $c_n = \frac{1}{n}$ D The sequence defined by $c_n = (-1)^n$ (oscillating)
Show answer & explanation

The sequence defined by $c_n = n$

The sequence $c_n = n$ diverges to infinity as $n \to \infty$.

Q17. What is the common ratio of the geometric series $\sum_{n=0}^{\infty} 3 \cdot (2^n)$?

Beginner
A 1 B 2 C 0.5 D 3
Show answer & explanation

2

The common ratio of the series is $2$, as each term is multiplied by $2$.

Building up · 6 questions

Q18. The series $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges if and only if:

Intermediate
A $p = 1$ B $p \ge 0$ C $p < 1$ D $p > 1$
Show answer & explanation

$p > 1$

This is the $p$-series test: it converges when $p > 1$ and diverges when $p \le 1$.

Q19. What is the sum of the geometric series $\sum_{n=0}^{\infty} \left(\frac{1}{3}\right)^n$?

Intermediate
A $\frac{3}{2}$ B 3 C $\frac{2}{3}$ D $\frac{1}{2}$
Show answer & explanation

$\frac{3}{2}$

For $|r|

Q20. If $\lim_{n \to \infty} a_n = 0$, what can be said about the series $\sum a_n$?

Intermediate
A It may converge or diverge; further tests are needed. B It converges to 0. C It always converges. D It always diverges.
Show answer & explanation

It may converge or diverge; further tests are needed.

The condition $\lim a_n = 0$ is necessary but not sufficient for convergence. For example, the harmonic series diverges even though $1/n \to 0$.

Q21. The alternating series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$ converges to:

Intermediate
A $\ln 2$ B $e$ C 1 D $\pi/4$
Show answer & explanation

$\ln 2$

This is the alternating harmonic series, which converges to $\ln 2$.

Q22. The series $\sum_{n=1}^{\infty} \frac{1}{n^2}$ is:

Intermediate
A Divergent by the $p$-series test B Convergent by the ratio test C Divergent by the $n$th term test D Convergent by the $p$-series test
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Convergent by the $p$-series test

This is a $p$-series with $p=2>1$, so it converges.

Q23. The integral test can be used to determine convergence of $\sum_{n=1}^{\infty} f(n)$ if $f(x)$ is:

Intermediate
A Positive and increasing for $x \ge 1$ B Continuous and oscillating C Differentiable and bounded D Positive, continuous, and decreasing for $x \ge 1$
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Positive, continuous, and decreasing for $x \ge 1$

The integral test requires $f$ to be positive, continuous, and decreasing on $[1,\infty)$.

Topper level · 2 questions

Q24. Which test is best suited to determine the convergence of $\sum_{n=1}^{\infty} \frac{n!}{n^n}$?

Advanced
A Integral test B Comparison test C Root test D Ratio test
Show answer & explanation

Ratio test

The ratio test is effective for series involving factorials and powers. Applying it shows convergence.

Q25. What is the radius of convergence of the power series $\sum_{n=0}^{\infty} \frac{x^n}{n!}$?

Advanced
A $\infty$ B 1 C e D 0
Show answer & explanation

$\infty$

Using the ratio test, the limit is 0 for all $x$, so the series converges for all real $x$; radius is infinite.

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