14. Sequences & Series

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]

Determine whether the given series is convergent using the Ratio Test.

${\sum }_{n=1}^{\infty }\frac{n^5}{{\left(6\right)}^n}$

Use the Alternating Series Test to approximate the sum of the series using the first five terms. 

${\sum }_{n=1}^{\infty }\frac{{(-1)}^{n+1}}{\sqrt{n}+2}$

Explain why the integral test does not apply to the series.

${\sum }_{n=0}^{\infty }\frac{n}{n^2-1}$

In Exercises 125–134, determine whether the sequence is monotonic, whether it is bounded, and whether it converges.

aₙ = (4ⁿ⁺¹ + 3ⁿ) / 4ⁿ

32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (from k = 1 to ∞) (−1)ᵏ k³ / √(k⁸ + 1)

Use the Direct Comparison Test to determine whether the series converges.

${\sum }_{n=1}^{\infty }\frac{3}{\sqrt{n}-2}$

Direct Comparison Test

In Exercises 1–8, use the Direct Comparison Test to determine if each series converges or diverges.

∑ (from n=1 to ∞) (√n + 1) / (√(n² + 3))

In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.

∑ (from n = 2 to ∞) [(ln n / n)³]

{Use of Tech} Bessel functions Bessel functions arise in the study of wave propagation in circular geometries (for example, waves on a circular drum head). They are conveniently defined as power series. One of an infinite family of Bessel functions is

J₀(x) = ∑ₖ₌₀∞ (−1)ᵏ/(2²ᵏ(k!)²) x²ᵏ

c. Differentiate J₀ twice and show (by keeping terms through x⁶) that J₀ satisfies the equation x² y′′(x) + xy′(x) + x²y(x)=0.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. Suppose 0 < aₖ < bₖ. If ∑ (k = 1 to ∞) aₖ converges, then ∑ (k = 1 to ∞) bₖ converges.

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)

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. A series that converges absolutely must converge.

40–62. Choose your test Use the test of your choice to determine whether the following series converge.

∑ (k = 2 to ∞) (5lnk) / k

Use the Limit Comparison Test to determine whether the series converges.

${\sum }_{n=1}^{\infty }\frac{n}{(n-1)3^{n+1}}$