Bn

Definition

The term 'bn' refers to the nth partial sum of an infinite series. It represents the sum of the first n terms of the series, which is used in the context of convergence and divergence tests, such as the Comparison Test covered in Section 5.4.

5 Must Know Facts For Your Next Test

1. The nth partial sum, $b_n$, is used to determine whether an infinite series converges or diverges when applying the Comparison Test.
2. The Comparison Test compares the behavior of the given series to the behavior of a known convergent or divergent series.
3. If the sequence of partial sums $\{b_n\}$ is bounded, then the series converges; if the sequence is unbounded, then the series diverges.
4. The Comparison Test is a powerful tool for determining the convergence or divergence of an infinite series when the direct comparison with a known series is possible.
5. The Comparison Test is particularly useful when the terms of the given series are difficult to evaluate or when the series does not fit the criteria of other convergence tests, such as the Integral Test or the Ratio Test.

Review Questions

• Explain the role of the nth partial sum, $b_n$, in the Comparison Test for infinite series.
  • The nth partial sum, $b_n$, plays a crucial role in the Comparison Test for infinite series. The Comparison Test compares the behavior of the given series to the behavior of a known convergent or divergent series. If the sequence of partial sums $\{b_n\}$ is bounded, then the series converges; if the sequence is unbounded, then the series diverges. By examining the behavior of the partial sums, the Comparison Test allows us to determine the convergence or divergence of the given infinite series.
• Describe how the Comparison Test is used to determine the convergence or divergence of an infinite series.
  • The Comparison Test is used to determine the convergence or divergence of an infinite series by comparing it to a known convergent or divergent series. To apply the Comparison Test, we first find a series with known convergence or divergence properties. We then compare the terms of the given series to the terms of the known series. If the terms of the given series are less than or equal to the terms of the known convergent series, then the given series converges. If the terms of the given series are greater than or equal to the terms of the known divergent series, then the given series diverges. By examining the behavior of the partial sums, $\{b_n\}$, the Comparison Test allows us to draw conclusions about the convergence or divergence of the infinite series.
• Analyze the significance of the Comparison Test in the context of infinite series and explain why it is a valuable tool for mathematicians.
  • The Comparison Test is a valuable tool in the study of infinite series because it allows mathematicians to determine the convergence or divergence of a series without having to evaluate the series directly. This is particularly useful when the terms of the given series are difficult to evaluate or when the series does not fit the criteria of other convergence tests. By comparing the behavior of the partial sums, $\{b_n\}$, of the given series to the partial sums of a known convergent or divergent series, the Comparison Test provides a way to draw conclusions about the convergence or divergence of the infinite series. This test is significant because it expands the repertoire of tools available to mathematicians for analyzing the behavior of infinite series, which is a fundamental concept in calculus and analysis.

Related terms

Partial Sum:

The partial sum of a series is the sum of the first n terms of the series, denoted as $s_n$ or $b_n$.

Infinite Series:

An infinite series is the sum of an infinite number of terms, where each term is a function of the index n.

Convergence:

Convergence refers to the property of an infinite series where the sequence of partial sums approaches a finite limit as the number of terms increases.