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If an and bn are convergent what can you deduce about lim an+k

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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.

User Niclas Von Caprivi
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