Question

Prove If an and bn are convergent lim(anᵏ) = (lim(an))ᵏ

Answer 1

To prove the statement, we need to show that the limit of the sequence of an^k is equal to the limit of the sequence an raised to the power of k. Assuming an and bn are convergent sequences, we can simplify the expression and take the limit to prove the statement.

Step-by-step explanation:

To prove the statement if an and bn are convergent, lim(ank) = (lim(an))k, we need to show that the limit of the sequence of ank is equal to the limit of the sequence an raised to the power of k.

Let's assume that an and bn are both convergent sequences. This means that liman = a and limbn = b.

Now, let's consider the sequence ank. Since an is convergent, we can write an = a. Then, taking the limit of both sides, we get lim(ank) = lim(ak) = ak. Therefore, lim(ank) = (lim(an))k.