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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.4.11
Chapter 10, Problem 10.4.11

9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.
∑ (k = 0 to ∞) 1 / (1000 + k)

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Identify the general term of the series, which is given by \(a_k = \frac{1}{1000 + k}\).
Recall the Divergence Test (also known as the nth-term test for divergence), which states: If \(\lim_{k \to \infty} a_k \neq 0\), then the series \(\sum a_k\) diverges. If the limit equals zero, the test is inconclusive.
Calculate the limit of the general term as \(k\) approaches infinity: \(\lim_{k \to \infty} \frac{1}{1000 + k}\).
Evaluate the limit by considering the behavior of the denominator as \(k\) becomes very large.
Based on the value of the limit, conclude whether the Divergence Test shows the series diverges or if the test is inconclusive.

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Key Concepts

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

Divergence Test

The Divergence Test states that if the limit of the terms of a series does not approach zero as k approaches infinity, then the series diverges. If the limit is zero, the test is inconclusive, and other methods must be used to determine convergence or divergence.
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Divergence Test (nth Term Test)

Behavior of the General Term

Analyzing the general term of the series, 1/(1000 + k), involves understanding its limit as k approaches infinity. Since the denominator grows without bound, the term approaches zero, which means the Divergence Test cannot confirm divergence in this case.
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Divergence Test (nth Term Test)

Comparison to Harmonic Series

The series ∑ 1/(1000 + k) resembles the harmonic series ∑ 1/k, which is known to diverge. Recognizing this similarity helps in concluding that despite terms approaching zero, the series diverges by comparison to a known divergent series.
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P-Series and Harmonic Series
Related Practice
Textbook Question

13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.


{bₙ}, where

bₙ = { n / (n + 1)if n ≤ 5000

ne⁻ⁿif n > 5000 }

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Textbook Question

11–27. Alternating Series Test Determine whether the following series converge.

∑ (k = 1 to ∞) (−1)ᵏ⁺¹ k¹/ᵏ

Textbook Question

21–42. Geometric series Evaluate each geometric series or state that it diverges.  


29.∑ (k = 1 to ∞) e^(–2k)

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Textbook Question

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 ∞) k¹⁰⁰ / (k + 1)!

Textbook Question

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.


∑ (k = 1 to ∞) (2 + (−1)ᵏ) / k²

Textbook Question

16–17. {Use of Tech} Periodic savings

Suppose you deposit m dollars at the beginning of every month in a savings account that earns a monthly interest rate of r, which is the annual interest rate divided by 12 (for example, if the annual interest rate is 2.4%, r = 0.024/12 = 0.002). For an initial investment of m dollars, the amount of money in your account at the beginning of the second month is the sum of your second deposit and your initial deposit plus interest, or m + m(1 + r). Continuing in this fashion, it can be shown that the amount of money in your account after n months is


Aₙ = m + m(1 + r) + ⋯ + m(1 + r)ⁿ⁻¹.


Use geometric sums to determine the amount of money in your savings account after 5 years (60 months) using the given monthly deposit and interest rate.


17. Monthly deposits of \$250 at a monthly interest rate of 0.2%

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