Problem 75 - Calculus: Early Transcendentals

Question

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? \( a_n = n(-1)^n \)

Step-by-Step Solution

Answer

The sequence is not monotonic and is unbounded.

Step 1: Define the Sequence

The sequence is defined as \( a_n = n(-1)^n \). This means each term \( a_n \) is obtained by multiplying the index \( n \) with \((-1)^n\).

Step 2: Determine the Pattern

Examine the sign of the terms for odd and even \( n \). For even \( n \), \((-1)^n = +1\), so \( a_n = n \). For odd \( n \), \((-1)^n = -1\), so \( a_n = -n \).

Step 3: Determine Monotonicity

To check if the sequence is monotonic, calculate the difference between consecutive terms. For consecutive terms, \( a_{n+1} - a_n = (n+1)(-1)^{n+1} - n (-1)^n \). This is not consistently positive or negative, hence the sequence is not monotonic.

Step 4: Determine Boundedness

The terms \( a_n = n \) (when \( n \) is even) increase indefinitely and the terms \( a_n = -n \) (when \( n \) is odd) decrease indefinitely as \( n \) increases. Thus, the sequence is unbounded.

Key Concepts

Monotonic Sequences

Monotonic sequences are sequences that consistently move in one direction, either increasing or decreasing, as they progress. An increasing sequence is one in which each term is greater than or equal to the term before it, while a decreasing sequence has each term less than or equal to the previous one. This characteristic can help predict behavior over time.
To determine if a sequence like \( a_n = n(-1)^n \) is monotonic, we must analyze the nature of its terms:

• If \( n \) is even, then \((-1)^n = 1\), making \( a_n = n \), which increases as \( n \) increases.
• If \( n \) is odd, then \((-1)^n = -1\), making \( a_n = -n \), which decreases as \( n \) increases.

We compute the difference between consecutive terms \( a_{n+1} - a_n \). For this sequence, this difference is not consistently positive or negative; hence, the sequence is not monotonic. A sequence that does not consistently follow an increasing or decreasing pattern is referred to as non-monotonic.

Bounded Sequences

A sequence is said to be bounded if there is a real number that serves as an upper or lower limit for all terms in the sequence. In simpler terms, a bounded sequence will not grow to infinity or decrease to negative infinity.
For the sequence \( a_n = n(-1)^n \), the terms behave differently depending on whether \( n \) is even or odd:

• For even \( n \), \( a_n = n \) which increases indefinitely without an upper bound.
• For odd \( n \), \( a_n = -n \) which decreases indefinitely without a lower bound.

Since the terms trend towards positive and negative infinity as \( n \) becomes larger or smaller, this sequence is unbounded. It does not have a single upper or lower limit, meaning it can take on arbitrarily large positive or negative values.

Alternating Sequences

Alternating sequences are a type of sequence characterized by their terms switching signs. This can lead to an oscillating pattern, often resulting in behavior that is quite different from non-alternating sequences.
In our sequence \( a_n = n(-1)^n \), each term alternates between positive and negative due to the presence of \((-1)^n\). Specifically:

• If \( n \) is even, \((-1)^n = 1\) and \( a_n \) is positive \((a_n = n)\).
• If \( n \) is odd, \((-1)^n = -1\) and \( a_n \) is negative \((a_n = -n)\).

This sign change at every step causes the sequence to oscillate between positive and negative values. Alternating sequences can be particularly useful in situations where balancing or canceling out elements is desirable, such as in certain summation problems.