Final Answer:
If aₙ and bₙ are convergent sequences, then the limit of their product, limₙ→∞ (aₙ ⋅ bₙ), is equal to the product of their limits, limₙ→∞ aₙ ⋅ limₙ→∞ bₙ.
Step-by-step explanation:
When aₙ and bₙ are convergent sequences, it means that as n approaches infinity, the terms of these sequences approach specific limits, denoted as Lₐ and L_b respectively. Mathematically, limₙ→∞ aₙ = Lₐ and limₙ→∞ bₙ = L_b.
Now, consider the product sequence cₙ = aₙ ⋅ bₙ. The limit of this product sequence is given by limₙ→∞ cₙ = limₙ→∞ (aₙ ⋅ bₙ). Using the properties of limits, we can express this as limₙ→∞ (aₙ ⋅ bₙ) = (limₙ→∞ aₙ) ⋅ (limₙ→∞ bₙ) = Lₐ ⋅ L_b.
This result follows from the fact that the limit of a product is equal to the product of the limits. Therefore, if aₙ and bₙ are convergent sequences, the limit of their product is the product of their limits. This property is fundamental in understanding the behavior of sequences and is widely used in mathematical analysis and calculus.