Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence boun

step 1 For this sequence, we'll actually show that it's not monotone and that it's also not bounded. No

Determine whether the sequence is increasing, decreasing, or...

Key Concepts

Alternating Sequences

Alternating sequences are sequences in which the signs of the terms alternate between positive and negative. This alternating behavior, often due to a negative base in an exponential expression, can prevent the sequence from being monotonic even if the magnitudes of the terms exhibit a consistent pattern.

Exponential Sequences

Exponential sequences are sequences in which each term is generated by raising a constant (the base) to a function of the index, often leading to rapid growth or decay. Their behavior is heavily influenced by the value of the base—if the base is greater than one in magnitude, the sequence tends to grow in absolute value, while a base with absolute value less than one results in decay.

Boundedness

A sequence is bounded if there exists a real number that serves as an upper bound and another that serves as a lower bound such that all terms of the sequence lie within these bounds. Determining boundedness is important as it informs us whether the sequence’s values remain within a fixed interval or if they escape to infinity.

Monotonicity

Monotonicity is a property of a sequence where its terms either never decrease (increasing sequence) or never increase (decreasing sequence). Analyzing a sequence for monotonicity involves examining the relationship between successive terms to determine if there is a consistent directional trend.

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