Skip to main content
Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblem 10.4.45
Chapter 10, Problem 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)

Verified step by step guidance
1
Recognize that the series is \( \sum_{n=1}^{\infty} \sin\left(\frac{1}{n}\right) \). Our goal is to determine whether this infinite series converges or diverges.
Recall the behavior of \( \sin x \) near zero: for small \( x \), \( \sin x \approx x \). Since \( \frac{1}{n} \to 0 \) as \( n \to \infty \), we can approximate \( \sin\left(\frac{1}{n}\right) \approx \frac{1}{n} \) for large \( n \).
Compare the given series to the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \), which is a well-known divergent series. Since \( \sin\left(\frac{1}{n}\right) \) behaves like \( \frac{1}{n} \) for large \( n \), the terms do not decrease fast enough to guarantee convergence.
Use the Limit Comparison Test by evaluating \( \lim_{n \to \infty} \frac{\sin(1/n)}{1/n} \). If this limit is a finite nonzero number, then both series either converge or diverge together.
Since the harmonic series diverges and the limit comparison test shows similar behavior, conclude that the series \( \sum_{n=1}^{\infty} \sin\left(\frac{1}{n}\right) \) diverges.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
6m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Convergence and Divergence of Infinite Series

An infinite series converges if the sum of its terms approaches a finite limit as the number of terms grows indefinitely; otherwise, it diverges. Understanding this concept is fundamental to analyzing whether a given series sums to a finite value or not.
Recommended video:
06:52
Convergence of an Infinite Series

Limit Comparison and Behavior of Terms

Examining the behavior of the terms as n approaches infinity helps determine convergence. If the terms do not approach zero, the series diverges. For example, since sin(1/n) ~ 1/n for large n, comparing with the harmonic series is useful.
Recommended video:
07:45
Limit Comparison Test

Comparison Test and Asymptotic Approximations

The Comparison Test involves comparing a given series to a known benchmark series to infer convergence or divergence. Using asymptotic approximations like sin(1/n) ≈ 1/n for large n allows applying this test effectively.
Recommended video:
09:25
Direct Comparison Test
Related Practice
Textbook Question

Recursively Defined Sequences

In Exercises 101–108, assume that each sequence converges and find its limit.

a₁ = 5,aₙ₊₁ = √(5aₙ)

Textbook Question

In Exercises 37–42, find the series’ radius of convergence.

∑ (from n = 1 to ∞) [ (n!)² / (2ⁿ (2n)!) ] xⁿ

Textbook Question

Finding nth Partial Sums

In Exercises 1–6, find a formula for the nth partial sum of each series and use it to find the series’ sum if the series converges.

2 + (2/3) + (2/9) + (2/27) + … + (2 / 3ⁿ⁻¹) + …

Textbook Question

In Exercises 35–40, find the first three nonzero terms of the Maclaurin series for each function.

f(x) = (sin x) ln(1 + x)

Textbook Question

In Exercises 81–86, find the values of x for which the given geometric series converges.

Also, find the sum of the series (as a function of x) for those values of x.

∑ (from n = 0 to ∞) [ (−1/2)ⁿ (x − 3)ⁿ ]

Textbook Question

Use power series operations to find the Taylor series at x = 0 for the functions in Exercises 13–30.

sin x – x + (x³ / 3!)