Textbook Question
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
1
views
Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 10, Problem 10.2.30
Repeating Decimals
Express each of the numbers in Exercises 23–30 as the ratio of two integers.
3.1̅4̅2̅8̅5̅7 = 3.142857142857 ...
Verified step by step guidance1
Identify the repeating decimal part of the number. Here, the decimal 3.1̅4̅2̅8̅5̅7̅ means the digits 142857 repeat indefinitely after the decimal point.
Let \( x = 3.142857142857... \). Since the repeating block has 6 digits, multiply both sides by \( 10^6 = 1,000,000 \) to shift the decimal point 6 places to the right: \( 10^6 x = 3142857.142857... \).
Subtract the original number \( x \) from this new equation to eliminate the repeating decimal part: \( 10^6 x - x = 3142857.142857... - 3.142857... \).
Simplify the subtraction to get \( (10^6 - 1) x = 3142854 \), which is \( 999999 x = 3142854 \).
Solve for \( x \) by dividing both sides by 999999: \( x = \frac{3142854}{999999} \). This fraction represents the repeating decimal as a ratio of two integers.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Repeating Decimals
A repeating decimal is a decimal number in which a sequence of digits repeats infinitely. For example, 3.142857142857... has the repeating block '142857'. Recognizing the repeating pattern is essential to convert the decimal into a fraction.
Recommended video:
08:23
Repeated Integration by Parts
Conversion of Repeating Decimals to Fractions
To express a repeating decimal as a ratio of two integers, set the decimal equal to a variable, multiply to shift the decimal point past the repeating block, and subtract to eliminate the repeating part. Solving the resulting equation yields the fraction form.
Recommended video:
11:26Partial Fraction Decomposition: Repeated Linear Factors
Properties of Rational Numbers
Rational numbers are numbers that can be expressed as the ratio of two integers. Every repeating decimal corresponds to a rational number, so understanding this property helps in confirming that the decimal can be converted into a fraction.
Recommended video:
06:21Properties of Functions
Related Practice
Textbook Question
Make up an infinite series of nonzero terms whose sum is
b. −3
Textbook Question
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
1
views
Textbook Question
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.)
Textbook Question
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
1
views
Textbook Question
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
1
views