Final answer:
The sum of two convergent sequences is also a convergent sequence, where the limit of the sum is equal to the sum of the individual limits of the sequences.
Step-by-step explanation:
If an and bn are convergent sequences, we can deduce that the limit of the sequence formed by the sum of these two, an + bn, is also convergent. According to the properties of limits, specifically the sum rule for limits, if the limit of an as n approaches infinity is L and the limit of bn as n approaches infinity is M, then the limit of the sum is L + M.
In other words, if:
Then:
- lim (an + bn) = lim an + lim bn = L + M
This outcome is guaranteed by the limit laws that state the limit of a sum is equal to the sum of the limits, provided that the limits exist and are finite.