Quiz: Infinite Series and Taylor Methods — 29 domande

1. What is an infinite sequence?

A. A function defined only on the real numbers

B. A sum of infinitely many numerical terms

C. An ordered list of numbers indexed by integers

D. A finite collection of unrelated numbers

Answer: (C)

Spiegazione

An infinite sequence is an ordered list of numbers, commonly indexed by the positive integers. A series, by contrast, is formed from sums of terms.

2. Which statement correctly describes convergence of a sequence?

A. Its terms eventually become exactly equal to zero

B. Its terms approach a finite number as the index grows

C. Its terms increase without bound as the index grows

D. Its terms alternate between two fixed values

Answer: (B)

Spiegazione

A sequence converges when its terms become arbitrarily close to a finite limit as the index approaches infinity. If no finite limit exists, the sequence diverges.

3. For which values of r does the sequence r^n converge?

A. For -1 ≤ r < 1

B. For every real value of r

C. For r ≥ 1

D. For -1 < r ≤ 1

Answer: (D)

Spiegazione

The sequence r^n converges precisely when -1 < r ≤ 1. Its limit is 0 for -1 < r < 1 and 1 when r = 1.

4. A decreasing sequence is bounded below. What conclusion follows?

A. It must be constant

B. It must diverge to negative infinity

C. It converges

D. It converges only if it is also bounded above

Answer: (C)

Spiegazione

Every bounded monotonic sequence converges. In particular, a decreasing sequence bounded below has a finite limit.

5. How is convergence of an infinite series defined?

A. Its terms approach a finite nonzero number

B. Its individual terms eventually become equal to zero

C. The sequence of its partial sums approaches a finite real number

D. The terms form a bounded ordered list

Answer: (C)

Spiegazione

An infinite series converges when the sequence of its partial sums approaches a finite real number. The terms themselves need not become exactly zero.

6. For the series with terms a_i, which expression gives its nth partial sum?

A. s_n = a_1a_2⋯a_n

B. s_n = a_n − a_1

C. s_n = Σ from i=1 to infinity of a_i

D. s_n = Σ from i=1 to n of a_i

Answer: (D)

Spiegazione

The nth partial sum is the finite sum s_n = Σ from i=1 to n of a_i. The infinite series is interpreted through the limit of these partial sums.

7. What is the sum of the geometric series Σ from n=1 to infinity of ar^(n−1) when |r| < 1?

A. ar/(1 − r)

B. 1/(a − r)

C. a/(1 + r)

D. a/(1 − r)

Answer: (D)

Spiegazione

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

8. What can be concluded about Σ a_n if lim a_n is nonzero or does not exist?

A. The series diverges

B. The series converges absolutely

C. The series converges to lim a_n

D. The series is guaranteed to be conditionally convergent

Answer: (A)

Spiegazione

If the terms of a series do not approach zero, the series diverges. When the terms do approach zero, the term test alone is inconclusive.

9. Under which conditions can the Integral Test compare Σ a_n with the improper integral of f?

A. f is bounded and periodic, with a_n = f(n)

B. f is discontinuous but nonnegative, with a_n = f(n)

C. f is continuous, positive, and decreasing on the relevant tail, with a_n = f(n)

D. f is differentiable, negative, and increasing, with a_n = f(n)

Answer: (C)

Spiegazione

The Integral Test requires f to be continuous, positive, and decreasing on the relevant tail. Under these conditions, the series and the improper integral have the same convergence behavior.

10. For which values of p does the p-series Σ from n=1 to infinity of 1/n^p converge?

A. p > 0

B. p > 1

C. p ≤ 1

D. p ≥ 1

Answer: (B)

Spiegazione

The p-series converges exactly when p > 1. The boundary case p = 1 is the harmonic series and diverges.

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

A. ∫ from n to infinity of f(x) dx ≤ R_n ≤ ∫ from n+1 to infinity of f(x) dx

B. R_n ≤ ∫ from 1 to n of f(x) dx

C. ∫ from n+1 to infinity of f(x) dx ≤ R_n ≤ ∫ from n to infinity of f(x) dx

D. R_n = ∫ from n to infinity of f(x) dx

Answer: (C)

Spiegazione

The integral remainder estimate places R_n between the integrals from n+1 and n to infinity. The lower bound uses the later starting point, while the upper bound uses n.

12. If 0 ≤ a_n ≤ b_n for every n and the series ∑b_n converges, what can be concluded about ∑a_n?

A. It oscillates without a limit

B. Its behavior cannot be determined

C. It converges

D. It diverges

Answer: (C)

Spiegazione

A nonnegative series bounded term by term above by a convergent series must also converge. The comparison provides an upper bound on its partial sums.

13. For positive-term series, what conclusion follows if lim(a_n/b_n) = c for some 0 < c < ∞?

A. The first series converges and the second diverges

B. The two series have the same individual terms

C. The limit comparison test cannot be applied

D. The two series either both converge or both diverge

Answer: (D)

Spiegazione

A positive finite ratio limit means the terms have comparable asymptotic size, so the two positive-term series share the same convergence behavior. A zero or infinite limit would not justify this conclusion.

14. Which condition is required for the alternating series ∑(-1)^n b_n to converge by the alternating series test?

A. The terms b_n increase and approach infinity

B. The terms b_n approach a nonzero finite value

C. The terms b_n decrease and approach zero

D. The signs alternate while b_n remains constant

Answer: (C)

Spiegazione

The alternating series test requires decreasing magnitudes b_n that tend to zero. Alternating signs alone do not guarantee convergence.

15. For an alternating series satisfying the alternating series test, how is the remainder R_n bounded?

A. |R_n| ≤ b_{n+1}

B. |R_n| ≤ b_n

C. |R_n| ≤ 1/b_{n+1}

D. |R_n| ≤ b_1 + b_{n+1}

Answer: (A)

Spiegazione

The alternating-series remainder is no larger in magnitude than the first omitted term, which is b_{n+1}. This bound depends on the alternating-series conditions.

16. Which statement correctly distinguishes absolute from conditional convergence?

A. Absolute convergence means ∑a_n converges, while conditional convergence means ∑|a_n| converges

B. Absolute convergence applies only to positive-term series, while conditional convergence applies only to negative-term series

C. Absolute and conditional convergence both require ∑|a_n| to diverge

D. Absolute convergence means ∑|a_n| converges, while conditional convergence means ∑a_n converges but ∑|a_n| diverges

Answer: (D)

Spiegazione

Absolute convergence occurs when the series of absolute values converges. Conditional convergence occurs when the original signed series converges but its absolute-value series diverges.

17. A series has terms that do not approach zero. What is the correct first conclusion when choosing a convergence test?

A. The series diverges immediately

B. The Ratio Test must be applied

C. The series must be rewritten as a geometric series

D. The series converges by comparison

Answer: (A)

Spiegazione

A necessary condition for convergence is that a_n approaches zero. If the term limit is nonzero or does not exist, the series diverges; a zero limit merely permits further testing.

18. Which test is most appropriately suggested for a series whose terms contain factorials or products?

A. The Root Test

B. The Ratio Test

C. The Integral Test

D. The Alternating Series Test

Answer: (B)

Spiegazione

The supplied method-selection guidance recommends the Ratio Test for factorials and products. It separately associates the Root Test with terms of the form (b_n)^n.

19. For which values of x does the geometric power-series identity \(\frac{1}{1-x}=\sum_{n=0}^{\infty}x^n\) hold directly?

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

B. For every real value of x

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

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

Answer: (A)

Spiegazione

The geometric identity converges when \(|x|<1\). Any endpoint must be checked separately after substitution because the identity does not automatically include \(|x|=1\).

20. When substituting an expression for x into a known power series, what must be transformed in addition to the series terms?

A. The order of differentiation

B. The convergence condition

C. The factorial coefficients only

D. The center of the function only

Answer: (B)

Spiegazione

Substitution changes both the terms of the series and the condition under which the resulting series converges. Changing only the terms gives an incomplete representation.

21. Inside its radius of convergence, how is the derivative of \(\sum_{n=0}^{\infty}c_n(x-a)^n\) obtained?

A. By differentiating each term to get \(\sum_{n=1}^{\infty}nc_n(x-a)^{n-1}\)

B. By differentiating only the first nonzero term

C. By replacing every power with \((x-a)^{n+1}\)

D. By integrating each term to get \(\sum_{n=1}^{\infty}\frac{c_n}{n}(x-a)^n\)

Answer: (A)

Spiegazione

Within the radius of convergence, term-by-term differentiation is valid and produces \(\sum_{n=1}^{\infty}nc_n(x-a)^{n-1}\).

22. For a Taylor series centered at a, what is the coefficient of \((x-a)^n\)?

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

B. \(f(a)/n!\)

C. \(f^{(n)}(0)/n!\)

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

Answer: (D)

Spiegazione

The coefficient of \((x-a)^n\) is \(c_n=f^{(n)}(a)/n!\). Derivatives evaluated at 0 specifically produce Maclaurin coefficients.

23. Which expression represents the nth Taylor polynomial for a function centered at a?

A. \(\sum_{i=0}^{n}\frac{f^{(i)}(a)}{i!}(x-a)^i\)

B. \(\sum_{i=0}^{\infty}\frac{f^{(i)}(a)}{i!}(x-a)^i\)

C. \(\sum_{i=0}^{n}f^{(i)}(a)i!(x-a)^i\)

D. \(\sum_{i=1}^{n}\frac{f^{(i)}(0)}{i!}x^i\)

Answer: (A)

Spiegazione

The nth Taylor polynomial is finite and includes terms from \(i=0\) through \(i=n\). The infinite version represents the Taylor series rather than the polynomial.

24. When does a formal Taylor series represent the function on an interval?

A. When the polynomial has a finite number of terms

B. When the remainder approaches zero on the interval

C. When the first derivative exists at the center

D. When the function has a nonzero value at the center

Answer: (B)

Spiegazione

A Taylor series equals the function only if the remainder \(R_n(x)=f(x)-T_n(x)\) approaches zero on the interval being considered.

25. Suppose \(|f^{(n+1)}(x)|\leq M\) whenever \(|x-a|\leq d\). What bound does Taylor’s Inequality give for the remainder?

A. \(|R_n(x)|\leq M(n+1)!|x-a|^{n+1}\)

B. \(|R_n(x)|\leq \frac{M|x-a|^{n+1}}{(n+1)!}\)

C. \(|R_n(x)|\leq \frac{M|x-a|^n}{n!}\)

D. \(|R_n(x)|\leq \frac{M}{|x-a|^{n+1}(n+1)!}\)

Answer: (B)

Spiegazione

Taylor’s Inequality bounds the remainder by \(M|x-a|^{n+1}/(n+1)!\) under the stated derivative bound.

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

A. 1

B. Infinity

C. 0

D. The distance from x to the origin

Answer: (B)

Spiegazione

The series \(e^x=\sum_{n=0}^{\infty}x^n/n!\) converges for every x, so its radius of convergence is infinite.

27. Which statement correctly distinguishes the standard Maclaurin series for sine and cosine?

A. Sine uses even powers, while cosine uses odd powers

B. Both sine and cosine use only odd powers

C. Sine uses odd powers, while cosine uses even powers

D. Both sine and cosine use only even powers

Answer: (C)

Spiegazione

The sine series contains powers \(x^{2n+1}\), which are odd, while the cosine series contains powers \(x^{2n}\), which are even. Both have infinite radius of convergence.

28. What is the correct sequence for constructing a Taylor polynomial approximation of degree n about a center a?

A. Choose the endpoint, compute only the first derivative, construct T_1, and evaluate the function at the center

B. Choose a, compute derivatives through order n, construct T_n, and bound the next derivative over the interval

C. Choose a, compute derivatives only at the endpoint, construct T_n, and ignore the remainder

D. Choose the midpoint, compute derivatives through order n+1, construct T_{n+1}, and bound the function itself

Answer: (B)

Spiegazione

Constructing T_n requires selecting the center, evaluating derivatives through order n, forming the polynomial, and bounding the next derivative over the entire interval. The approximation is local to the chosen center, so error control requires attention to the interval.

29. Which method is appropriate for estimating the error of a Taylor approximation when a bound for the relevant derivative is available?

A. Apply Taylor's Inequality using the derivative bound

B. Evaluate the Taylor polynomial only at its center

C. Use the linear formula regardless of the polynomial's degree

D. Replace the remainder by the next Taylor coefficient without checking its size

Answer: (A)

Spiegazione

Taylor's Inequality estimates the remainder when a suitable bound on a derivative is known over the interval. Evaluating only at the center does not measure the approximation error away from that point.