Final answer:
When the sequences an and bn are convergent, adding a constant k to an does not affect its convergence, and the new sequence an + k will also converge to the original limit plus k.
Step-by-step explanation:
If an and bn are convergent sequences, then we can deduce some properties about the sequence an + k, where k is a constant.
Specifically, since an is convergent, it means that there exists a limit L such that as n approaches infinity, the sequence an approaches L.
Given the definition of convergence, when we add a constant k to every term of this convergent sequence, the resulting sequence an + k will also be convergent. The limit of this new sequence will be L + k, because adding a constant to each term of a convergent sequence shifts the limit by the same constant.
To deduce this formally, we can write:
lim (an + k) as n approaches infinity equals lim (an) + lim (k) as n approaches infinity.
Since lim (an) as n approaches infinity equals L (the limit of the original sequences), and lim (k) as n approaches infinity is just k.
Therefore, lim (an + k) as n approaches infinity equals L + k.
This illustrates that the convergence of a sequence is preserved under the addition of a constant.