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

검증된 단계별 안내

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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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.

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