Make up an infinite series of nonzero terms whose sum is
b. −3

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Recall that an infinite series is the sum of infinitely many terms, and for the series to have a finite sum, it must be convergent.

One common type of infinite series with a known sum is a geometric series, which has the form \(\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}\), where \(|r| < 1\).

To create a series that sums to \(b - 3\), choose the first term \(a\) and common ratio \(r\) such that \(\frac{a}{1-r} = b - 3\).

For example, pick a value for \(r\) with \(|r| < 1\), then solve for \(a = (b - 3)(1 - r)\) to ensure the sum matches \(b - 3\).

Write the infinite series explicitly as \(\sum_{n=0}^{\infty} a r^n\), substituting the values of \(a\) and \(r\) you chose, which will be a series of nonzero terms summing to \(b - 3\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. For the series to have a finite sum, it must converge, meaning its partial sums approach a specific value. Understanding convergence criteria is essential to construct a series that sums to a given number.

Geometric Series

A geometric series has terms that multiply by a constant ratio each time. If the absolute value of the ratio is less than one, the series converges to a sum given by a/(1 - r), where a is the first term and r is the ratio. This formula helps create series with a desired sum.

Constructing Series with a Given Sum

To create an infinite series summing to a specific value, choose terms that satisfy convergence and sum conditions. For example, adjusting the first term and ratio in a geometric series allows the sum to equal any real number, including negative values like −3.

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