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Problem 10.2.89
Make up an infinite series of nonzero terms whose sum is
b. −3
Problem 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)
Problem 10.2.19
In Exercises 15–22, determine if the geometric series converges or diverges. If a series converges, find its sum.
1 − (2/e) + (2/e)² − (2/e)³ + (2/e)⁴ − …
Problem 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)
Problem 10.1.15
Finding a Sequence’s Formula
In Exercises 13–30, find a formula for the nth term of the sequence.
1, -4, 9, -16, 25, … Squares of the positive integers, with alternating signs
Problem 10.5.10
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 ∞) [4ⁿ / (3n)ⁿ]
Problem 10.1.123
In Exercises 121–124, determine whether the sequence is monotonic and whether it is bounded.
aₙ = 2ⁿ 3ⁿ / n!
Problem 10.9.2
Finding Taylor Series
Use substitution (as in Formula (7)) to find the Taylor series at x = 0 of the functions in Exercises 1–12.
e⁻ˣ/²
Problem 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
Problem 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.)
Problem 10.9.45
Error Estimates
The approximation eˣ = 1 + x + (x² / 2) is used when x is small. Use the Remainder Estimation Theorem to estimate the error when |x| < 0.1.
Problem 10.6.94
Does the series
∑ (from n=1 to ∞) (1/n − 1/n²)
converge or diverge? Justify your answer.
Problem 10.7.40
In Exercises 37–42, find the series’ radius of convergence.
∑ (from n = 1 to ∞) [ n! xⁿ / nⁿ ]
Problem 10.2.64
Which series in Exercises 53–76 converge, and which diverge? Give reasons for your answers. If a series converges, find its sum.
∑ (from n = 1 to ∞) (1 − 1/n)ⁿ
Problem 10.3.2
Applying the Integral Test
Use the Integral Test to determine if the series in Exercises 1–12 converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.
∑ (from n = 1 to ∞) 1 / n⁰·²
Problem 10.3.28
Determining Convergence or Divergence
Which of the series in Exercises 13–46 converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)
∑ (from n=1 to ∞) (1 + 1/n)ⁿ
Problem 10.4.69
In Exercises 67–72, use the results of Exercises 63 and 64 to determine if each series converges or diverges.
∑(from n=2 to ∞) [(ln n)¹⁰⁰⁰ / n¹.⁰⁰¹]
Problem 10.8.10
Finding Taylor Polynomials
In Exercises 1–10, find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a.
f(x) = √(1 − x), a = 0
Problem 10.8.25
Finding Taylor and Maclaurin Series
In Exercises 25–34, find the Taylor series generated by f at x = a.
f(x) = x³ − 2x + 4, a = 2
Problem 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)]
Problem 10.1.139
Is it true that a sequence {aₙ} of positive numbers must converge if it is bounded above? Give reasons for your answer.
Problem 10.1.81
Convergence and Divergence
Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = (xⁿ / (2n + 1))^(1/n), x > 0
Problem 10.8.2
Finding Taylor Polynomials
In Exercises 1–10, find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a.
f(x) = sin x, a = 0
Problem 10.1.105
Recursively Defined Sequences
In Exercises 101–108, assume that each sequence converges and find its limit.
a₁ = 5, aₙ₊₁ = √(5aₙ)
Problem 10.7.10
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 = 1 to ∞) [ (x − 1)ⁿ / √n ]
Problem 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]
Problem 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
Problem 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
Problem 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
Problem 10.10.29
Use series to evaluate the limits in Exercises 29–40.
29. lim (x → 0) (e^x - (1 + x)) / x²