Ch. 10 - Infinite Sequences and Series

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 10 - Infinite Sequences and Series

Problem 10.1.139

Chapter 10, Problem 10.1.139

Is it true that a sequence {aₙ} of positive numbers must converge if it is bounded above? Give reasons for your answer.

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Recall the definition of convergence for a sequence: a sequence \( \{a_n\} \) converges if there exists a limit \( L \) such that for every \( \epsilon > 0 \), there is an \( N \) where for all \( n > N \), \( |a_n - L| < \epsilon \).

Understand that being bounded above means there exists some number \( M \) such that \( a_n \leq M \) for all \( n \), but this does not guarantee the sequence approaches a specific value.

Consider that a sequence can be bounded above but still oscillate or fail to settle down to a single limit, for example, a sequence that jumps between two values less than \( M \).

Note that for convergence, the sequence must get arbitrarily close to one particular number, not just stay below a certain bound.

Conclude that a sequence of positive numbers bounded above does not necessarily converge; boundedness alone is not sufficient to ensure convergence.

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Key Concepts

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

Bounded Sequences

A sequence is bounded above if there exists a number that is greater than or equal to every term in the sequence. Being bounded means the terms do not grow beyond a certain limit, but it does not guarantee convergence by itself.

Convergence of Sequences

A sequence converges if its terms approach a specific finite limit as the index goes to infinity. Convergence requires the terms to get arbitrarily close to a single value, not just to be bounded.

Monotone Convergence Theorem

This theorem states that every bounded monotone sequence converges. If a sequence is both bounded and monotone (either non-increasing or non-decreasing), it must converge, but boundedness alone is insufficient.

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Related Practice

Textbook Question

In Exercises 37–42, find the series’ radius of convergence.

∑ (from n = 1 to ∞) [ ( n / (n + 1) )ⁿ^ ² ] xⁿ (Hint: Apply the Root Test.)

Textbook Question

Which series in Exercises 53–76 converge, and which diverge? Give reasons for your answers. If a series converges, find its sum.

∑ (from n = 1 to ∞) (1 − 1/n)ⁿ

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Textbook Question

Intervals of Convergence

In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?

∑ (from n = 0 to ∞) (x + 5)ⁿ

Textbook Question

Finding Taylor Polynomials

In Exercises 1–10, find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a.

f(x) = sin x,a = 0

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Textbook Question

Finding Taylor Series

Use substitution (as in Formula (7)) to find the Taylor series at x = 0 of the functions in Exercises 1–12.

e⁻ˣ/²

Textbook Question

Use series to evaluate the limits in Exercises 29–40.

29. lim (x → 0) (e^x - (1 + x)) / x²