What is the sum of the series [math: (−1/3)0+(−1/3)1+(−1/3)2+(−1/3)3+…](-1/3)^0+(-1/3)^1+(-1/3)^2+(-1/3)^3+\ldots Remember to enter your answer in lowest terms.

Question

BlackTom AI · Accepted Answer

The problem presents an infinite geometric series with terms formed as (-1/3)^n for n = 0, 1, 2, ... .

First, identify the key parameters of a geometric series: the first term a, and the common ratio r between successive terms. Here, the first term is a = (-1/3)^0 = 1, and the common ratio is r = -1/3 since each term multiplies by -1/3 to get the next term.

Next, check convergence. An infinite geometric series converges if |r| < 1. Since |r| = |-1/3| = 1/3 < 1, the series does converge and we can use the standard sum formula for an infinite geometric series.

Apply the sum formula S = a / (1 - r). Substituting a = 1 and r = -1/3, we get S = 1 / (1 - (-1/3)) = 1 / (1 + 1/3) = 1 / (4/3).

Compute the division: dividing by a fraction is the same as multiplying by its reciprocal, so S = 1 × (3/4) = 3/4.

Finally, this result is already in lowest terms because 3/4 cannot be reduced further. The steps above show how the sum is derived from the geometric-series framework, using the first term and the common ratio and confirming convergence before applying the formula.

What is the sum of the series [math: (−1/3)0+(−1/3)1+(−1/3)2+(−1/3)3+…](-1/3)^0+(-1/3)^1+(-1/3)^2+(-1/3)^3+\ldots Remember to enter your answer in lowest terms.

View Explanation

Log in for full answers

Similar Questions

Question12 Consider the series [math]. Which of the following correctly says what the series is equal to on its interval of convergence?Select one alternative: [math] [math] [math] [math] ResetMaximum marks: 1 Flag question undefined

What is the first term of the geometric series with and ? (type number only into box)

What would be the sum of the first 18 terms of the geometric series with: 20 1.4 (type answer only in box correct to the nearest whole number)

What is the sum of the series \[(-1/3)^0+(-1/3)^1+(-1/3)^2+(-1/3)^3+\ldots\] Remember to enter your answer in lowest terms.

(K:1) Find S8 for the series 3+15+75+375+…

The sum to infinity of a geometric sequence is 20 and the common ratio r is 0.2. What are the first four terms of the sequence?

The sum of the first seven terms of the geometric sequence: 0.2, 0.8, 3.2, … is closest to:

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler