Question

For what values of $r$ is the sequence $\left\{n r^2\right\}$ convergent?

    For what values of $r$ is the sequence $\left\{n r^2\right\}$ convergent?

Calculus

James Stewart 7th Edition

Chapter 11, Problem 69

Instant Answer

Solved by Expert J Hardin

03/13/2018

Step 1

The sequence given is \(\{n r^2\}\), where \(n\) is a natural number and \(r\) is a real number. We need to determine for which values of \(r\) this sequence converges. Show more…

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For what values of $r$ is the sequence $\left\{n r^2\right\}$ convergent?

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Key Concepts

Sequence Convergence

This concept relates to whether the terms of a sequence approach a specific finite value as the sequence progresses. A sequence is said to converge if, for every arbitrarily small positive tolerance, there is a stage in the sequence beyond which all terms differ from the limit by less than that tolerance.

Unbounded Growth

When the terms of a sequence increase without bound—that is, they eventually exceed any fixed number—the sequence diverges. In sequences where each term is the product of an unbounded index and a non-zero constant, the terms tend to grow indefinitely, leading to divergence.

Role of Constant Factors in Sequences

Multiplying the index of a sequence by a constant factor affects its convergence behavior. If the constant is zero, the product yields a constant sequence, which converges. However, if the constant is non-zero, even a linear multiplication can cause the sequence to become unbounded as the index increases.

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