Skip to main content
Ch. 8 - Sequences, Induction, and Probability
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 8 - Sequences, Induction, and ProbabilityProblema 43
Capitolo 9, Problema 43

Find the sum of each infinite geometric series. ∑i=1∞8(−0.3)i−1\(\sum\)_{i=1}^{\(\infty\)} 8(-0.3)^{i-1}

Guida verificata passo dopo passo
1
Identify the first term \( a \) of the infinite geometric series. Since the series is given by \( \sum_{i=1}^{\infty} 8(-0.3)^{i-1} \), the first term corresponds to \( i=1 \), so \( a = 8(-0.3)^{0} = 8 \).
Determine the common ratio \( r \) of the geometric series. The common ratio is the factor by which each term is multiplied to get the next term, which is \( r = -0.3 \).
Check the condition for the sum of an infinite geometric series to exist. The sum converges only if the absolute value of the common ratio is less than 1, i.e., \( |r| < 1 \). Since \( |-0.3| = 0.3 < 1 \), the series converges.
Use the formula for the sum of an infinite geometric series: \[ S = \frac{a}{1 - r} \]. Substitute the values of \( a = 8 \) and \( r = -0.3 \) into the formula.
Simplify the expression \( \frac{8}{1 - (-0.3)} \) to find the sum of the series. This will give you the sum without calculating the final numeric value.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
6m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Infinite Geometric Series

An infinite geometric series is the sum of infinitely many terms where each term is found by multiplying the previous term by a constant ratio. It converges only if the absolute value of the common ratio is less than 1, allowing the sum to approach a finite value.
Video consigliato:
4:18
Geometric Sequences - Recursive Formula

Common Ratio

The common ratio in a geometric series is the factor by which each term is multiplied to get the next term. It is crucial for determining whether the series converges and for calculating the sum of the series when |r| < 1.
Video consigliato:
5:57
Graphs of Common Functions

Sum Formula for Infinite Geometric Series

The sum of an infinite geometric series with first term a and common ratio r (where |r| < 1) is given by S = a / (1 - r). This formula allows us to find the finite sum of the series when it converges.
Video consigliato:
4:18
Geometric Sequences - Recursive Formula
Pratica correlata
Domanda del libro di testo

Find the term indicated in each expansion. (x2 + y3)8; sixth term

603
views
Domanda del libro di testo

Find the sum of the even integers between 21 and 45.

1281
views
Domanda del libro di testo

In Exercises 45-46, it is equally probable that the pointer on the spinner shown will land on any one of the eight regions, numbered 1 through 8. If the pointer lands on a borderline, spin again. Find the probability that the pointer will stop on an odd number or a number less than 6.

1152
views
Domanda del libro di testo

Express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 12+22+32+⋯+1521^2+2^2+3^2+⋯+ 15^2

869
views
Domanda del libro di testo

Use the formula for the sum of the first n terms of a geometric sequence to find the indicated sum.

∑i=165i\(\sum\)_{i=1}^{6} 5^i

1170
views
Domanda del libro di testo

Find the sum of the odd integers between 30 and 54.

707
views