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Chapter 7: Problem 36

Restate the symbolic version of the formula for evaluating a geometric series using summation notation.

Short Answer

Expert verified
The symbolic formula for evaluating a geometric series using summation notation can be restated as: \(S_n = \sum_{i=1}^{n} a_i = \sum_{i=1}^{n} a_1 * r^{i-1} = \frac{a_1(1 - r^n)}{1 - r}\)

Step by step solution

01

Identify the general formula for a geometric series

The formula for evaluating a geometric series is given by: \(S_n = \frac{a_1(1 - r^n)}{1 - r}\) where \(S_n\) is the sum of the first \(n\) terms of the series, \(a_1\) is the first term, \(r\) is the common ratio between consecutive terms, and \(n\) is the number of terms.
02

Identify the general term of a geometric series

In a geometric series, the general term is given by: \(a_n = a_1 * r^{n-1}\) where \(a_n\) is the \(n^{th}\) term of the series, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the position of the term in the series.
03

Convert the general term into summation notation

To represent the geometric series using summation notation, we use the general term and sum it from the first term to the last term: \(\sum_{i=1}^{n} a_i = \sum_{i=1}^{n} a_1 * r^{i-1}\) Notice that the variable \(i\) in the summation takes care of the position of each term in the series.
04

Restate the formula for evaluating a geometric series using summation notation

Finally, we can restate the symbolic formula for evaluating a geometric series using summation notation by combining the formula for the sum of the series and the summation notation: \(S_n = \sum_{i=1}^{n} a_i = \sum_{i=1}^{n} a_1 * r^{i-1} = \frac{a_1(1 - r^n)}{1 - r}\)

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Most popular questions from this chapter

Explain why the polynomial factorization $$ 1-x^{n}=(1-x)\left(1+x+x^{2}+\cdots+x^{n-1}\right) $$ holds for every integer \(n \geq 2\).

Consider a geometric sequence with first term \(b\) and ratio \(r\) of consecutive terms. (a) Write the sequence using the three-dot notation, giving the first four terms. (b) Give the \(100^{\text {th }}\) term of the sequence. \(b=2, r=\frac{1}{2}\)

Consider the sequence whose \(n^{\text {th }}\) term \(a_{n}\) is given by the indicated formula. (a) Write the sequence using the three-dot notation, giving the first four terms of the sequence. (b) Write the sequence as a recursive sequence. \(a_{n}=2^{n} n !\)

Evaluate \(\lim _{n \rightarrow \infty}\left(1-\frac{1}{n}\right)^{n}\)

Evaluate \(\lim _{n \rightarrow \infty} \frac{4 n-2}{7 n+6}\)
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