QCM : Infinite Sequences and Series — 27 questions

Question 1

1. What distinguishes a sequence from a series?

A sequence must converge, whereas a series must diverge

A sequence uses positive terms, whereas a series uses negative terms

A sequence is an ordered list of terms, whereas a series is a sum of terms

A sequence is a sum of terms, whereas a series is an ordered list of terms

Explication

A sequence is an ordered list of numbers, with each term indexed by a positive integer. A series is formed by adding the terms of a sequence.

Answer

A sequence is an ordered list of terms, whereas a series is a sum of terms

Question 2

2. What does the statement $a_n\to L$ as $n\to\infty$ mean?

The terms increase without bound and therefore approach $L$

The terms become equal to $L$ after a fixed finite number of steps

The terms become arbitrarily close to the finite number $L$ for sufficiently large $n$

The terms alternate between $L$ and values increasingly far from $L$

Explication

The notation means that, for sufficiently large indices, the terms can be made arbitrarily close to $L$. Because $L$ is finite, the sequence converges.

Answer

The terms become arbitrarily close to the finite number $L$ for sufficiently large $n$

Question 3

3. For which values of $r$ does the sequence $\{r^n\}$ converge?

For $r<-1$ or $r\geq1$

For $|r|>1$

For $-1\leq r<1$

For $-1<r\leq1$

Explication

The sequence $r^n$ converges precisely when $-1<r\leq1$. Its limit is $0$ when $-1<r<1$ and $1$ when $r=1$.

Answer

For $-1<r\leq1$

Question 4

4. A sequence is increasing and bounded above. What conclusion follows from the Monotonic Sequence Theorem?

The sequence diverges to negative infinity

The sequence must alternate between two values

The sequence converges

The sequence has no finite upper bound

Explication

An increasing sequence bounded above is bounded and monotonic, so the Monotonic Sequence Theorem guarantees that it converges.

Answer

The sequence converges

Question 5

5. How is convergence of an infinite series $\sum_{n=1}^{\infty}a_n$ defined?

Its partial sums increase without bound

Its terms approach a finite real number

Its individual terms eventually become exactly zero

Its partial sums approach a finite real number

Explication

A series converges when its partial sums $s_n=\sum_{i=1}^n a_i$ approach a finite real number. This differs from convergence of the individual sequence of terms.

Answer

Its partial sums approach a finite real number

Question 6

6. What is the sum of the geometric series $\sum_{n=1}^{\infty}ar^{n-1}$ when $|r|<1$?

$\frac{r}{1-a}$

$\frac{a}{1-r}$

$\frac{1-r}{a}$

$\frac{a}{1+r}$

Explication

A geometric series with first term $a$ and common ratio $r$ converges for $|r|<1$, and its sum is $a/(1-r)$.

Answer

$\frac{a}{1-r}$

Question 7

7. What conclusion follows if $\lim_{n\to\infty}a_n$ does not exist or is not zero?

The series can converge only if its partial sums are increasing

The series $\sum a_n$ converges to zero

The sequence $\{a_n\}$ must be geometric

The series $\sum a_n$ diverges

Explication

The Test for Divergence states that a series must have terms approaching zero to converge. If the limit does not exist or is not zero, the series diverges.

Answer

The series $\sum a_n$ diverges

Question 8

8. Which conditions are required to apply the Integral Test to $\sum a_n$ with $a_n=f(n)$?

$f$ must be discontinuous, positive, and increasing on the relevant tail

$f$ must be bounded above but need not be positive or decreasing

$f$ must be continuous, positive, and decreasing on the relevant tail

$f$ must be continuous and alternating on the relevant tail

Explication

The Integral Test requires that $f$ be continuous, positive, and decreasing on the relevant tail. Under these conditions, the series and the improper integral either both converge or both diverge.

Answer

$f$ must be continuous, positive, and decreasing on the relevant tail

Question 9

9. For which values of $p$ does the p-series $\sum_{n=1}^{\infty}\frac{1}{n^p}$ converge?

$p\geq1$

$p\leq1$

$p>1$

$p<1$

Explication

The p-series converges exactly when $p>1$ and diverges when $p\leq1$. In particular, the boundary case $p=1$ is the divergent harmonic series.

Answer

$\int_{n+1}^{\infty}f(x)\,dx\leq R_n\leq\int_n^{\infty}f(x)\,dx$

Answer

Its terms are bounded above by those of a convergent positive series

Answer

They have the same convergence behavior

Answer

The magnitudes decrease and approach zero

Answer

Absolute convergence means $\sum|a_n|$ converges, while conditional convergence means $\sum a_n$ converges but $\sum|a_n|$ diverges

Answer

Conclude that the series diverges

Answer

Factorials or products—Ratio Test; whole nth powers—Root Test

Answer

For $|x|<1$

Answer

It remains the same

Answer

$\frac{f^{(n)}(a)}{n!}$

Answer

A Taylor series centered at zero

Answer

The polynomial is a finite partial sum

Answer

When the remainder approaches zero on that interval

Answer

For $|x|<1$

Answer

f(a)+f'(a)(x-a)

Answer

It must bound |f⁽ⁿ⁺¹⁾(x)| throughout the relevant interval.

Question 10

10. For a convergent positive decreasing series with $a_n=f(n)$, which inequality gives bounds for the remainder $R_n$?

$\int_n^{\infty}f(x)\,dx\leq R_n\leq\int_{n+1}^{\infty}f(x)\,dx$

$\int_{n}^{n+1}f(x)\,dx\leq R_n\leq\int_{1}^{n}f(x)\,dx$

$\int_{n+1}^{\infty}f(x)\,dx\leq R_n\leq\int_n^{\infty}f(x)\,dx$

$R_n\leq\int_{1}^{n}f(x)\,dx\leq R_n$

Explication

For a positive decreasing series, the remainder is bounded below by the integral beginning at $n+1$ and above by the integral beginning at $n$.

Question 11

11. Which condition allows Direct Comparison to prove that a positive-term series converges?

Its terms are larger than those of a convergent positive series

Its terms are bounded above by those of a convergent positive series

Its terms have a ratio of zero to those of a convergent series

Its terms alternate while remaining larger than a convergent series

Explication

If 0 ≤ a_n ≤ b_n and the series ∑b_n converges, then ∑a_n also converges. Being larger than a convergent series does not establish convergence.

Question 12

12. Suppose positive terms satisfy $\lim_{n\to\infty} a_n/b_n=3$. What can be concluded about the series $\sum a_n$ and $\sum b_n$?

They have the same convergence behavior

The first converges and the second diverges

The comparison is invalid because the limit is not 1

The first diverges and the second converges

Explication

Because the limit is finite and strictly positive, Limit Comparison applies and the two positive-term series either both converge or both diverge.

Question 13

13. Under which conditions does the Alternating Series Test establish convergence of $\sum(-1)^n b_n$?

The magnitudes decrease but approach a positive constant

The magnitudes decrease and approach zero

The magnitudes increase and approach zero

The signs alternate and the magnitudes approach infinity

Explication

The Alternating Series Test requires the sequence b_n to decrease and satisfy b_n → 0. Alternating signs alone are not sufficient.

Question 14

14. An alternating series satisfies the conditions of the Alternating Series Test. If its first omitted term has magnitude $b_{n+1}$, what is the maximum possible magnitude of the remainder?

$2b_{n+1}$

$b_n$

The sum of all omitted magnitudes

$b_{n+1}$

Explication

For a qualifying alternating series, the remainder satisfies $|R_n|\leq b_{n+1}$. Thus the first omitted term provides an upper bound for the error.

Question 15

15. Which statement correctly distinguishes absolute convergence from conditional convergence?

Absolute convergence means $\sum|a_n|$ converges, while conditional convergence means $\sum a_n$ converges but $\sum|a_n|$ diverges

Absolute convergence means $\sum a_n$ diverges while $\sum|a_n|$ converges

Conditional convergence means $\sum|a_n|$ and $\sum a_n$ have identical convergence behavior

Absolute convergence means both $\sum a_n$ and $\sum|a_n|$ diverge

Explication

A series is absolutely convergent when its series of absolute values converges. It is conditionally convergent when the original series converges but the absolute-value series diverges.

Question 16

16. A series has terms that do not approach zero. What should happen when selecting a convergence test?

Use the Ratio Test because the terms are nonzero

Apply the Alternating Series Test immediately

Conclude that the series diverges

Compare it with a geometric series before deciding

Explication

A necessary condition for convergence is that a_n → 0. If the term limit is nonzero, the series diverges, so no further convergence test is needed.

Question 17

17. Which pairing best matches the form of a series with a useful convergence test?

Rational p-series terms—Ratio Test; alternating signs—Root Test

Factorials or products—Ratio Test; whole nth powers—Root Test

Geometric terms—Root Test; factorials—Direct Comparison

Factorials or products—Root Test; whole nth powers—Ratio Test

Explication

The Ratio Test is particularly useful for factorials and products, while the Root Test is suited to terms of the form $(b_n)^n$.

Question 18

18. For which values of x is the geometric power-series identity $\frac{1}{1-x}=\sum_{n=0}^{\infty}x^n$ justified?

For $|x|>1$

For every real value of x

For $|x|<1$

For $|x|\leq 1$

Explication

The geometric series converges to $1/(1-x)$ when $|x|<1$. Outside this interval, the formal infinite sum is not justified by the geometric identity.

Question 19

19. What happens to the radius of convergence when a power series is differentiated or integrated term by term?

It always doubles

It remains the same

It becomes infinite

It becomes zero

Explication

Within the original radius of convergence, term-by-term differentiation or integration produces a series with the same radius of convergence.

Question 20

20. What is the coefficient of $(x-a)^n$ in the Taylor series of $f$ centered at $a$?

$\frac{f(a)}{n!}$

$\frac{f^{(n)}(0)}{n!}$

$\frac{f^{(n)}(a)}{n!}$

$f^{(n)}(a)n!$

Explication

For a Taylor series centered at $a$, the coefficient of $(x-a)^n$ is the nth derivative evaluated at $a$, divided by $n!$. Derivatives at zero are used specifically for Maclaurin series.

Question 21

21. Which description correctly identifies a Maclaurin series?

A power series with no constant term

A Taylor series centered at one

A finite Taylor polynomial of degree zero

A Taylor series centered at zero

Explication

A Maclaurin series is the special case of a Taylor series centered at $a=0$, so it is written in powers of $x$.

Question 22

22. What distinguishes the nth-degree Taylor polynomial from the full Taylor series?

The polynomial has infinitely many terms

The polynomial is a finite partial sum

The polynomial always equals the function

The polynomial uses derivatives at zero instead of at a

Explication

The nth-degree Taylor polynomial contains only the terms through degree $n$, making it a finite partial sum. It need not equal the function it approximates.

Question 23

23. Under what condition does a function equal its Taylor series on an interval?

When the center of the series lies outside the interval

When the function has a nonzero derivative

When the Taylor polynomial has degree one

When the remainder approaches zero on that interval

Explication

The function equals its Taylor series where the remainder $R_n(x)=f(x)-T_n(x)$ approaches zero as the degree increases.

Question 24

24. What is the radius of convergence of the Maclaurin series for $e^x$?

One

Infinite

Zero

It depends on x

Explication

The exponential series $e^x=\sum_{n=0}^{\infty}x^n/n!$ converges for every real x, so its radius of convergence is infinite. This differs from the geometric series, whose radius is one.

Question 25

25. For which values of x is the binomial series $(1+x)^k=\sum_{n=0}^{\infty}\binom{k}{n}x^n$ stated to hold?

For $|x|<1$

For every real value of x

For $|x|>1$

For $|x|\leq 1$

Explication

The binomial series representation is given for $|x|<1$, with coefficients $\binom{k}{n}=\frac{k(k-1)\cdots(k-n+1)}{n!}$.

Question 26

26. Which expression gives the linear Taylor approximation to a function f centered at a?

f(a)+f'(a)(x-a)

f'(a)+f(a)(x-a)

f(a)+f''(a)(x-a)^2

f(a)+f'(a)(x-a)+f''(a)(x-a)^2/2

Explication

The degree-one Taylor polynomial is T₁(x)=f(a)+f'(a)(x-a). Terms involving higher derivatives belong to higher-degree Taylor polynomials.

Question 27

27. To apply Taylor’s Inequality for an nth-degree Taylor polynomial, what must the constant M satisfy?

It must equal |f⁽ⁿ⁺¹⁾(a)| at the center point.

It must equal the maximum value of |f(x)| on the interval.

It must bound |f⁽ⁿ⁾(x)| only at the endpoint.

It must bound |f⁽ⁿ⁺¹⁾(x)| throughout the relevant interval.

Explication

Taylor’s Inequality requires |f⁽ⁿ⁺¹⁾(x)|≤M on the entire relevant interval. A derivative value at just the center point is not sufficient to establish the remainder bound.

QCM : Infinite Sequences and Series — 27 questions

Questions et réponses du QCM

1. What distinguishes a sequence from a series?

2. What does the statement $a_n\to L$ as $n\to\infty$ mean?

3. For which values of $r$ does the sequence $\{r^n\}$ converge?

4. A sequence is increasing and bounded above. What conclusion follows from the Monotonic Sequence Theorem?

5. How is convergence of an infinite series $\sum_{n=1}^{\infty}a_n$ defined?

6. What is the sum of the geometric series $\sum_{n=1}^{\infty}ar^{n-1}$ when $|r|<1$?

7. What conclusion follows if $\lim_{n\to\infty}a_n$ does not exist or is not zero?

8. Which conditions are required to apply the Integral Test to $\sum a_n$ with $a_n=f(n)$?

9. For which values of $p$ does the p-series $\sum_{n=1}^{\infty}\frac{1}{n^p}$ converge?

10. For a convergent positive decreasing series with $a_n=f(n)$, which inequality gives bounds for the remainder $R_n$?

11. Which condition allows Direct Comparison to prove that a positive-term series converges?

12. Suppose positive terms satisfy $\lim_{n\to\infty} a_n/b_n=3$. What can be concluded about the series $\sum a_n$ and $\sum b_n$?

13. Under which conditions does the Alternating Series Test establish convergence of $\sum(-1)^n b_n$?

14. An alternating series satisfies the conditions of the Alternating Series Test. If its first omitted term has magnitude $b_{n+1}$, what is the maximum possible magnitude of the remainder?

15. Which statement correctly distinguishes absolute convergence from conditional convergence?

16. A series has terms that do not approach zero. What should happen when selecting a convergence test?

17. Which pairing best matches the form of a series with a useful convergence test?

18. For which values of x is the geometric power-series identity \(\frac{1}{1-x}=\sum_{n=0}^{\infty}x^n\) justified?

19. What happens to the radius of convergence when a power series is differentiated or integrated term by term?

20. What is the coefficient of \((x-a)^n\) in the Taylor series of \(f\) centered at \(a\)?

21. Which description correctly identifies a Maclaurin series?

22. What distinguishes the nth-degree Taylor polynomial from the full Taylor series?

23. Under what condition does a function equal its Taylor series on an interval?

24. What is the radius of convergence of the Maclaurin series for \(e^x\)?

25. For which values of x is the binomial series \((1+x)^k=\sum_{n=0}^{\infty}\binom{k}{n}x^n\) stated to hold?

26. Which expression gives the linear Taylor approximation to a function f centered at a?

27. To apply Taylor’s Inequality for an nth-degree Taylor polynomial, what must the constant M satisfy?

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