Explanation Result used: Let ∑ n = 0 ∞ a 1 r n be a geometric series of terms, the partial sums S n represent the sum first n terms of the series, that is S n = a 1 + a 1 r + a 1 r 2 + … + a 1 r n − 1 . The series ∑ n = 0 ∞ a 1 r n is convergent when | r | < 1 and divergent when | r | ≥ 1 . The sum of the series S = ∑ n = 0 ∞ a 1 r n = a 1 1 − r , when | r | < 1 . Calculation: Consider the decimal 0.151515 … . The above decimal can be rewritten as follows. 0.151515 … = 0.15 + 0.0015 + 0.000015 + … = 15 × 10 − 2 + 15 × 10 − 4 + 15 × 10 − 6 + … = 15 10 2 + 15 10 4 + 15 10 6 + … = 15 100 + 15 10 , 000 + 15 1 , 000 , 000 + … Therefore, the decimal 0.151515 … can be expressed as, 15 100 + 15 10 , 000 + 15 1 , 000 , 000 + … . Here, the first term of the geometric series can be written as 15 100 ( 1 100 ) 0 , the second term is, 15 100 ( 1 100 ) 1 and the third term of the geometric series is, 15 100 ( 1 100 ) 2 and so on. Therefore, the terms of the geometric series can be rewritten as, 15 100 ( 1 100 ) 0 + 15 100 ( 1 100 ) 1 + 15 100 ( 1 100 ) 2 + ⋯

11th Edition

ISBN: 9780134437705

Author: Washington

Publisher: PEARSON

See similar textbooks

Videos

Question

Chapter 30.1, Problem 39E

To determine

To express: The repeating decimal 0.151515… as 15100+1510,000+151,000,000+… and find the sum of the series.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 30.1, Problem 38E

Chapter 30.1, Problem 40E

Students have asked these similar questions

(^) k Recall that for numbers 0 ≤ k ≤ n the binomial coefficient (^) is defined as n! k! (n−k)! Question 1. (1) Prove the following identity: (22) + (1121) = (n+1). (2) Use the identity above to prove the binomial theorem by induction. That is, prove that for any a, b = R, n (a + b)" = Σ (^) an- n-kyk. k=0 n Recall that Σ0 x is short hand notation for the expression x0+x1+ +xn- (3) Fix x = R, x > 0. Prove Bernoulli's inequality: (1+x)" ≥1+nx, by using the binomial theorem. - Question 2. Prove that ||x| - |y|| ≤ |x − y| for any real numbers x, y. Question 3. Assume (In) nEN is a sequence which is unbounded above. That is, the set {xn|nЄN} is unbounded above. Prove that there are natural numbers N] k for all k Є N. be natural numbers (nk Є N). Prove that

Question content area top Part 1 Find the measure of ABC for the congruent triangles ABC and Upper A prime Upper B prime Upper C primeA′B′C′. 79 degrees79° 1533 2930 Part 1 m ABCequals=enter your response heredegrees

Joy is making Christmas gifts. She has 6 1/12 feet of yarn and will need 4 1/4 to complete our project. How much yarn will she have left over compute this solution in two different ways

Chapter 30 Solutions

Basic Technical Mathematics

Ch. 30.1 - Prob. 1PE Ch. 30.1 - Prob. 2PE Ch. 30.1 - Prob. 1E Ch. 30.1 - Prob. 2E Ch. 30.1 - Prob. 3E Ch. 30.1 - Prob. 4E Ch. 30.1 - Prob. 5E Ch. 30.1 - Prob. 6E Ch. 30.1 - Prob. 7E Ch. 30.1 - Prob. 8E

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

To express: The repeating decimal 0.151515 … as 15 100 + 15 10 , 000 + 15 1 , 000 , 000 + … and find the sum of the series.

Videos

To express: The repeating decimal 0.151515… as 15100+1510,000+151,000,000+… and find the sum of the series.

Want to see the full answer?

Chapter 30 Solutions