Σ. (-1) 23 4 n = 0 = Π

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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a) graph the sequence of partial sums
b) use the graph to estimate the sum of the series.
(please handwrite work not type last time i couldn't make heads or tails of whoever gave the typed solution)

The given summation is an infinite series represented as follows:

\[
\sum_{n=0}^{\infty} \frac{23}{4} \left( -\frac{11}{12} \right)^n
\]

This expression includes:

1. **Summation** (\(\sum\)): The symbol \(\sum\) indicates that the terms in the series are to be added together from \(n = 0\) to infinity (\(\infty\)).

2. **Fraction \(\frac{23}{4}\)**: The term \(\frac{23}{4}\) is a constant multiplier for each term in the series.

3. **Power of a Fraction** \(\left( -\frac{11}{12} \right)^n\): This is a geometric series where each term involves raising the fraction \(-\frac{11}{12}\) to the power of \(n\).

This is a classic geometric series where the initial term \(a = \frac{23}{4}\) and the common ratio \(r = -\frac{11}{12}\). For convergence, the absolute value of the common ratio must be less than 1. Here, \(|-\frac{11}{12}| = \frac{11}{12}\), which is less than 1, so the series converges.

The sum \(S\) of an infinite geometric series can be calculated using the formula:

\[
S = \frac{a}{1 - r}
\]

where \(a\) is the first term and \(r\) is the common ratio.
Transcribed Image Text:The given summation is an infinite series represented as follows: \[ \sum_{n=0}^{\infty} \frac{23}{4} \left( -\frac{11}{12} \right)^n \] This expression includes: 1. **Summation** (\(\sum\)): The symbol \(\sum\) indicates that the terms in the series are to be added together from \(n = 0\) to infinity (\(\infty\)). 2. **Fraction \(\frac{23}{4}\)**: The term \(\frac{23}{4}\) is a constant multiplier for each term in the series. 3. **Power of a Fraction** \(\left( -\frac{11}{12} \right)^n\): This is a geometric series where each term involves raising the fraction \(-\frac{11}{12}\) to the power of \(n\). This is a classic geometric series where the initial term \(a = \frac{23}{4}\) and the common ratio \(r = -\frac{11}{12}\). For convergence, the absolute value of the common ratio must be less than 1. Here, \(|-\frac{11}{12}| = \frac{11}{12}\), which is less than 1, so the series converges. The sum \(S\) of an infinite geometric series can be calculated using the formula: \[ S = \frac{a}{1 - r} \] where \(a\) is the first term and \(r\) is the common ratio.
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