Textbook Question
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(3ⁿ⁺¹ + 3)⁄3ⁿ}
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14. Sequences & Series
Sequences
1:35 minutesProblem 10.2.55
Textbook Question
55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(−1)ⁿ / 2ⁿ}
Verified step by step guidance1
Identify the general term of the sequence, which is given by \(a_n = \frac{(-1)^n}{2^n}\).
Recall that \((-1)^n\) alternates between \$1\( and \)-1\( as \)n$ increases, causing the numerator to alternate signs.
Note that the denominator \$2^n\( grows exponentially as \)n$ increases, becoming very large.
Consider the behavior of the absolute value of the terms: \(\left|a_n\right| = \frac{1}{2^n}\), which approaches \$0$ as \(n \to \infty\).
Since the numerator only changes sign but the magnitude approaches zero, conclude that the sequence converges to \$0$.
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1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sequence and Limit
A sequence is an ordered list of numbers defined by a specific formula. The limit of a sequence is the value that the terms approach as the index goes to infinity. Understanding how to find limits helps determine if a sequence converges to a finite value or diverges.
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Introduction to Sequences
Behavior of Exponential Terms
Exponential terms like 2ⁿ grow rapidly as n increases. When in the denominator, such terms cause the overall fraction to approach zero. Recognizing this behavior is key to evaluating limits involving exponential expressions.
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Alternating Sequences
An alternating sequence changes sign with each term, often represented by (−1)ⁿ. While the sign alternates, the magnitude may approach zero or another value. Understanding this helps analyze whether the sequence converges or oscillates without settling.
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