뒤로
문제 10.4.62
Suppose that aₙ > 0 and limₙ→∞ n²aₙ = 0. Prove that ∑aₙ converges.
문제 10.1.22
Finding a Sequence’s Formula
In Exercises 13–30, find a formula for the nth term of the sequence.
2, 6, 10, 14, 18, … Every other even positive integer
문제 10.7.50
In Exercises 43–50, use Theorem 20 to find the series’ interval of convergence and, within this interval, the sum of the series as a function of x.
∑ (from n = 0 to ∞) [ (x² − 1) / 2 ]ⁿ
문제 10.8.18
Finding Taylor Series at x = 0 (Maclaurin Series)
Find the Maclaurin series for the functions in Exercises 11–24.
5 cos πx
문제 10.9.16
Use power series operations to find the Taylor series at x = 0 for the functions in Exercises 13–30.
sin x – x + (x³ / 3!)
문제 10.2.89
Make up an infinite series of nonzero terms whose sum is
b. −3
문제 10.4.17
Determining Convergence or Divergence
Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.
∑ (from n=1 to ∞) 1 / (2√n + ³√n)
문제 10.1.4
Finding Terms of a Sequence
Each of Exercises 1–6 gives a formula for the nth term aₙ of a sequence {aₙ}. Find the values of a₁, a₂, a₃, and a₄.
aₙ = 2 + (-1)ⁿ
문제 10.6.10
Determining Convergence or Divergence
In Exercises 1–14, determine whether the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.
∑ (from n = 2 to ∞) [(-1)ⁿ⁺¹ (1 / ln n)]
문제 10.6.26
Absolute and Conditional Convergence
Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.
∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (ⁿ√10)]
문제 10.6.1
Determining Convergence or Divergence
In Exercises 1–14, determine whether the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.
∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (1 / n^(3/2))]
문제 10.1.99
Convergence and Divergence
Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = (1/n) ∫₁ⁿ (1/x) dx
문제 10.1.36
Convergence and Divergence
Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = (n + 3) / (n² + 5n + 6)
문제 10.2.59
Which series in Exercises 53–76 converge, and which diverge? Give reasons for your answers. If a series converges, find its sum.
∑ (from n = 0 to ∞) e^(−2n)
문제 10.1.73
Convergence and Divergence
Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = (2n + 2)! / (2n − 1)!
문제 10.6.94
Does the series
∑ (from n=1 to ∞) (1/n − 1/n²)
converge or diverge? Justify your answer.
문제 10.7.6
Intervals of Convergence
In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?
∑ (from n = 0 to ∞) (2x)ⁿ
문제 10.6.66
In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑ (from n = 0 to ∞) [((n + 1) / (n + 2))ⁿ]
문제 10.1.123
In Exercises 121–124, determine whether the sequence is monotonic and whether it is bounded.
aₙ = 2ⁿ 3ⁿ / n!
문제 10.10.1
Find the first four nonzero terms of the Taylor series for the functions in Exercises 1–10.
6. (1 - x/3)^4
문제 10.5.2
Using the Ratio Test
In Exercises 1–8, use the Ratio Test to determine whether each series converges absolutely or diverges.
∑(from n=1 to ∞) [(-1)ⁿ (n + 2) / 3ⁿ]
문제 10.8.33
Finding Taylor and Maclaurin Series
In Exercises 25–34, find the Taylor series generated by f at x = a.
f(x) = cos(2x + π/2), a = π/4
문제 10.7.42
In Exercises 37–42, find the series’ radius of convergence.
∑ (from n = 1 to ∞) [ ( n / (n + 1) )ⁿ^ ² ] xⁿ (Hint: Apply the Root Test.)
문제 10.6.80
In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑ (from n = 0 to ∞) [(2n + 3)(2ⁿ + 3) / (3ⁿ + 2)]
문제 10.5.23
Determining Convergence or Divergence
In Exercises 17–46, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑(from n=1 to ∞) [(2 + (−1)ⁿ) / 1.25ⁿ]
문제 10.3.64
Use the Integral Test to show that the series
∑ (from n=0 to ∞) e^(−n²)
converges.
문제 10.5.15
Using the Root Test
In Exercises 9–16, use the Root Test to determine if each series converges absolutely or diverges.
∑(from n=1 to ∞) [(-1)ⁿ (1 − 1/n)ⁿ^²]
(Hint: lim (n→∞) (1 + x/n)ⁿ = eˣ)
문제 10.4.15
Limit Comparison Test
In Exercises 9–16, use the Limit Comparison Test to determine if each series converges or diverges.
∑ (from n=2 to ∞) 1 / ln n
(Hint: Limit Comparison with ∑ (from n=2 to ∞) (1/n))
문제 10.10.29
Use series to evaluate the limits in Exercises 29–40.
29. lim (x → 0) (e^x - (1 + x)) / x²
문제 10.4.45
Determining Convergence or Divergence
Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.
∑ (from n=1 to ∞) sin (1/n)