Cuestionario: Infinite Series and Taylor Methods — 29 preguntas

Question 1

1. What is an infinite sequence?

A function defined only on the real numbers

A sum of infinitely many numerical terms

An ordered list of numbers indexed by integers

A finite collection of unrelated numbers

Explicación

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.

Answer

An ordered list of numbers indexed by integers

Question 2

2. Which statement correctly describes convergence of a sequence?

Its terms eventually become exactly equal to zero

Its terms approach a finite number as the index grows

Its terms increase without bound as the index grows

Its terms alternate between two fixed values

Explicación

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.

Answer

Its terms approach a finite number as the index grows

Question 3

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

For -1 ≤ r < 1

For every real value of r

For r ≥ 1

For -1 < r ≤ 1

Explicación

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

Answer

For -1 < r ≤ 1

Question 4

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

It must be constant

It must diverge to negative infinity

It converges

It converges only if it is also bounded above

Explicación

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

Answer

It converges

Question 5

5. How is convergence of an infinite series defined?

Its terms approach a finite nonzero number

Its individual terms eventually become equal to zero

The sequence of its partial sums approaches a finite real number

The terms form a bounded ordered list

Explicación

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

Answer

The sequence of its partial sums approaches a finite real number

Question 6

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

s_n = a_1a_2⋯a_n

s_n = a_n − a_1

s_n = Σ from i=1 to infinity of a_i

s_n = Σ from i=1 to n of a_i

Explicación

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.

Answer

s_n = Σ from i=1 to n of a_i

Question 7

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

ar/(1 − r)

1/(a − r)

a/(1 + r)

a/(1 − r)

Explicación

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

Answer

a/(1 − r)

Question 8

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

The series diverges

The series converges absolutely

The series converges to lim a_n

The series is guaranteed to be conditionally convergent

Explicación

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.

Answer

The series diverges

Question 9

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

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

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

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

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

Explicación

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.

Answer

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

Question 10

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

p > 0

p > 1

p ≤ 1

p ≥ 1

Explicación

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

Answer

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

Answer

It converges

Answer

The two series either both converge or both diverge

Answer

The terms b_n decrease and approach zero

Answer

|R_n| ≤ b_{n+1}

Answer

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

Answer

The series diverges immediately

Answer

The Ratio Test

Answer

For \(|x|<1\)

Answer

The convergence condition

Answer

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

Answer

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

Answer

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

Answer

When the remainder approaches zero on the interval

Answer

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

Answer

Sine uses odd powers, while cosine uses even powers

Answer

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

Answer

Apply Taylor's Inequality using the derivative bound

Question 11

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

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

R_n ≤ ∫ from 1 to n of f(x) dx

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

R_n = ∫ from n to infinity of f(x) dx

Explicación

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.

Question 12

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

It oscillates without a limit

Its behavior cannot be determined

It converges

It diverges

Explicación

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.

Question 13

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

The first series converges and the second diverges

The two series have the same individual terms

The limit comparison test cannot be applied

The two series either both converge or both diverge

Explicación

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.

Question 14

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

The terms b_n increase and approach infinity

The terms b_n approach a nonzero finite value

The terms b_n decrease and approach zero

The signs alternate while b_n remains constant

Explicación

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

Question 15

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

|R_n| ≤ b_{n+1}

|R_n| ≤ b_n

|R_n| ≤ 1/b_{n+1}

|R_n| ≤ b_1 + b_{n+1}

Explicación

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.

Question 16

16. Which statement correctly distinguishes absolute from conditional convergence?

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

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

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

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

Explicación

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.

Question 17

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

The series diverges immediately

The Ratio Test must be applied

The series must be rewritten as a geometric series

The series converges by comparison

Explicación

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.

Question 18

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

The Root Test

The Ratio Test

The Integral Test

The Alternating Series Test

Explicación

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.

Question 19

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

For \(|x|<1\)

For every real value of x

For \(|x|>1\)

For \(|x|\leq 1\)

Explicación

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

Question 20

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

The order of differentiation

The convergence condition

The factorial coefficients only

The center of the function only

Explicación

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.

Question 21

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

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

By differentiating only the first nonzero term

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

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

Explicación

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

Question 22

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

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

\(f(a)/n!\)

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

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

Explicación

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

Question 23

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

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

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

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

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

Explicación

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.

Question 24

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

When the polynomial has a finite number of terms

When the remainder approaches zero on the interval

When the first derivative exists at the center

When the function has a nonzero value at the center

Explicación

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.

Question 25

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

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

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

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

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

Explicación

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

Question 26

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

1

Infinity

0

The distance from x to the origin

Explicación

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

Question 27

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

Sine uses even powers, while cosine uses odd powers

Both sine and cosine use only odd powers

Sine uses odd powers, while cosine uses even powers

Both sine and cosine use only even powers

Explicación

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.

Question 28

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

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

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

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

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

Explicación

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.

Question 29

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

Apply Taylor's Inequality using the derivative bound

Evaluate the Taylor polynomial only at its center

Use the linear formula regardless of the polynomial's degree

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

Explicación

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.

Cuestionario: Infinite Series and Taylor Methods — 29 preguntas

Preguntas y respuestas detalladas

1. What is an infinite sequence?

2. Which statement correctly describes convergence of a sequence?

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

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

5. How is convergence of an infinite series defined?

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

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

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

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

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

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

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

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

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

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

16. Which statement correctly distinguishes absolute from conditional convergence?

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

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

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

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

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

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

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

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

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

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

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

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

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

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