Infinite Sequences and Series
8 minFull explanation, worked examples & memory tricks in the app
Kit Library / Mathematics / Calculus
⚡ Topic Learning KitSequences & Series
English 25 leveled MCQs Free
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All 25 questions, laddered from warm-up to topper-level, each with its answer and the reasoning behind it.
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
Answers and explanations are in the Veda app.
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
$\sum_{n=1}^{\infty} \frac{1}{n}$
Answers and explanations are in the Veda app.
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
Answers and explanations are in the Veda app.
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 $\sum a_n = L$.
Answers and explanations are in the Veda app.
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
$a_n = n$
Answers and explanations are in the Veda app.
Q6. Every convergent sequence is bounded.
Beginner
A
True
B
False
Answers and explanations are in the Veda app.
Q7. If a series converges, then its terms must approach zero.
Beginner
A
True
B
False
Answers and explanations are in the Veda app.
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
Answers and explanations are in the Veda app.
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 $n \to \infty$
Answers and explanations are in the Veda app.
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
$S_n = a + n \cdot r$
Answers and explanations are in the Veda app.
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
Answers and explanations are in the Veda app.
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
Answers and explanations are in the Veda app.
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
$\sum_{n=1}^{\infty} \frac{1}{n^3}$
Answers and explanations are in the Veda app.
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
$a_n = a + nd$
Answers and explanations are in the Veda app.
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 $p < 0$
Answers and explanations are in the Veda app.
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)
Answers and explanations are in the Veda app.
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
Answers and explanations are in the Veda app.
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
$p > 1$
Answers and explanations are in the Veda app.
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
$\frac{1}{2}$
Answers and explanations are in the Veda app.
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.
Answers and explanations are in the Veda app.
Q21. The alternating series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$ converges to:
Intermediate
A
$\ln 2$
B
$e$
C
1
D
$\pi/4$
Answers and explanations are in the Veda app.
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
Answers and explanations are in the Veda app.
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 $x \ge 1$
Answers and explanations are in the Veda app.
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
Answers and explanations are in the Veda app.
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
Answers and explanations are in the Veda app.
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