Quiz: Infinite Sequences and Series — 27 Fragen

1. What distinguishes a sequence from a series?

A. A sequence must converge, whereas a series must diverge

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

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

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

Answer: (C)

Erklärung

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.

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

A. The terms increase without bound and therefore approach $L$

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

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

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

Answer: (C)

Erklärung

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

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

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

B. For $|r|>1$

C. For $-1\leq r<1$

D. For $-1<r\leq1$

Answer: (D)

Erklärung

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

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

A. The sequence diverges to negative infinity

B. The sequence must alternate between two values

C. The sequence converges

D. The sequence has no finite upper bound

Answer: (C)

Erklärung

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

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

A. Its partial sums increase without bound

B. Its terms approach a finite real number

C. Its individual terms eventually become exactly zero

D. Its partial sums approach a finite real number

Answer: (D)

Erklärung

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.

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

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

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

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

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

Answer: (B)

Erklärung

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

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

A. The series can converge only if its partial sums are increasing

B. The series $\sum a_n$ converges to zero

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

D. The series $\sum a_n$ diverges

Answer: (D)

Erklärung

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.

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

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

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

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

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

Answer: (C)

Erklärung

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.

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

A. $p\geq1$

B. $p\leq1$

C. $p>1$

D. $p<1$

Answer: (C)

Erklärung

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.

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

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

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

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

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

Answer: (C)

Erklärung

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

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

A. Its terms are larger than those of a convergent positive series

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

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

D. Its terms alternate while remaining larger than a convergent series

Answer: (B)

Erklärung

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.

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 $\sum b_n$?

A. They have the same convergence behavior

B. The first converges and the second diverges

C. The comparison is invalid because the limit is not 1

D. The first diverges and the second converges

Answer: (A)

Erklärung

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

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

A. The magnitudes decrease but approach a positive constant

B. The magnitudes decrease and approach zero

C. The magnitudes increase and approach zero

D. The signs alternate and the magnitudes approach infinity

Answer: (B)

Erklärung

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

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?

A. $2b_{n+1}$

B. $b_n$

C. The sum of all omitted magnitudes

D. $b_{n+1}$

Answer: (D)

Erklärung

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.

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

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

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

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

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

Answer: (A)

Erklärung

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.

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

A. Use the Ratio Test because the terms are nonzero

B. Apply the Alternating Series Test immediately

C. Conclude that the series diverges

D. Compare it with a geometric series before deciding

Answer: (C)

Erklärung

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.

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

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

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

C. Geometric terms—Root Test; factorials—Direct Comparison

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

Answer: (B)

Erklärung

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

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

A. For \(|x|>1\)

B. For every real value of x

C. For \(|x|<1\)

D. For \(|x|\leq 1\)

Answer: (C)

Erklärung

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.

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

A. It always doubles

B. It remains the same

C. It becomes infinite

D. It becomes zero

Answer: (B)

Erklärung

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

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

A. \(\frac{f(a)}{n!}\)

B. \(\frac{f^{(n)}(0)}{n!}\)

C. \(\frac{f^{(n)}(a)}{n!}\)

D. \(f^{(n)}(a)n!\)

Answer: (C)

Erklärung

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.

21. Which description correctly identifies a Maclaurin series?

A. A power series with no constant term

B. A Taylor series centered at one

C. A finite Taylor polynomial of degree zero

D. A Taylor series centered at zero

Answer: (D)

Erklärung

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

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

A. The polynomial has infinitely many terms

B. The polynomial is a finite partial sum

C. The polynomial always equals the function

D. The polynomial uses derivatives at zero instead of at a

Answer: (B)

Erklärung

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.

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

A. When the center of the series lies outside the interval

B. When the function has a nonzero derivative

C. When the Taylor polynomial has degree one

D. When the remainder approaches zero on that interval

Answer: (D)

Erklärung

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

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

A. One

B. Infinite

C. Zero

D. It depends on x

Answer: (B)

Erklärung

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.

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?

A. For \(|x|<1\)

B. For every real value of x

C. For \(|x|>1\)

D. For \(|x|\leq 1\)

Answer: (A)

Erklärung

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

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

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

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

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

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

Answer: (A)

Erklärung

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

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

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

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

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

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

Answer: (D)

Erklärung

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.