Final Answer:
If both aₙ and bₙ are convergent sequences, the limit of aₙ to the power of k (lim(aₙ)ᵏ) will also converge, assuming k is a constant.
Step-by-step explanation:
When aₙ and bₙ are convergent sequences, it means that as n approaches infinity, both sequences approach a common limit. Let lim(aₙ) = A and lim(bₙ) = B. Now, if we consider the sequence aₙᵏ, where k is a constant, the limit of this sequence can be expressed as lim(aₙ)ᵏ = Aᵏ. This is a result of the continuity of exponentiation. In other words, taking a convergent sequence to a constant power preserves convergence, and the limit of the resulting sequence is the original limit raised to the power of the constant.
For example, let aₙ = (1/n), which converges to 0 as n approaches infinity. If we take k = 2, then lim(aₙ)² = 0, and the sequence is still convergent. However, if k were a variable dependent on n, the convergence might not be guaranteed, and further analysis would be required.
Understanding the behavior of convergent sequences and the impact of operations, such as exponentiation, on their limits is fundamental in mathematical analysis. It allows for the prediction of the behavior of sequences and is widely applicable in various mathematical and scientific disciplines.