Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problem 10.1.7

Chapter 10, Problem 10.1.7

The first ten terms of the sequence {(1 + 1/10ⁿ)^10ⁿ}∞ ₙ₌₁ are rounded to 8 digits right of the decimal point (see table). Make a conjecture about the limit of the sequence.
n an
1 2.59374246
2 2.70481383
3 2.71692393
4 2.71814593
5 2.71826824
6 2.71828047
7 2.71828169
8 2.71828179
9 2.71828204
10 2.71828203

Verified step by step guidance

Recognize that the given sequence is defined as $a_n = \left(1 + \frac{1}{10^n}\right)^{10^n}$, where $n$ is a positive integer.

Recall the well-known limit definition of the mathematical constant $e$, which is $\lim_{m \to \infty} \left(1 + \frac{1}{m}\right)^m = e$.

Notice that in the sequence, the exponent and the denominator inside the parentheses are both powers of 10, specifically \$10^n$, which grows very large as $n$ increases.

Based on the pattern of the terms given and the known limit definition, conjecture that as $n$ approaches infinity, $a_n$ approaches the constant $e$.

To confirm this conjecture, you could compare the numerical values of $a_n$ for large $n$ with the known decimal expansion of $e \approx 2.718281828\ldots$ and observe the convergence.

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Key Concepts

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

Sequences and Limits

A sequence is an ordered list of numbers defined by a specific rule. The limit of a sequence is the value that the terms approach as the index goes to infinity. Understanding limits helps determine the long-term behavior of sequences, which is essential for making conjectures about convergence.

Exponential Functions and the Number e

The expression (1 + 1/n)^n is a classic sequence that approaches the mathematical constant e (~2.71828) as n becomes large. This sequence illustrates how exponential growth can be approximated by limits, and recognizing this helps identify the limit in the given problem.

Rounding and Numerical Approximation

Rounding numbers to a fixed number of decimal places affects the precision of numerical values. When analyzing sequences numerically, understanding rounding helps interpret the data correctly and assess how close the terms are to the conjectured limit.

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Related Practice

Textbook Question

45–48. {Use of Tech} Explicit formulas for sequences Consider the formulas for the following sequences {aₙ}ₙ₌₁∞

Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges.

aₙ = ⁿ² + n ;n = 1, 2, 3, …

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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 ∞) 2ᵏ k! / kᵏ

Textbook Question

54–69. Telescoping series

For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.

65. ∑ (k = 1 to ∞) (1 / √(k + 1) – 1 / √(k + 3))

Textbook Question

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from j = 2 to ∞)1 / (j ln¹⁰j)

Textbook Question

13–52. Limits of sequences

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

{√(n² + 1) − n}

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Textbook Question

13–52. Limits of sequences

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

{2ⁿ⁺¹3⁻ⁿ}