Determine whether the sequence converges or diverges. Justify.
1) an = ((3n)2 )/(3n + 1)
To determine whether the sequence converges or diverges, we need to find its limit as n approaches infinity.
We can simplify the expression inside the sequence by dividing both the numerator and denominator by n:
an = ((3n)2 )/(3n + 1)
= (9n^2)/(3n + 1)
As n approaches infinity, the term n + 1 becomes negligible compared to 3n. Thus, we can ignore the "+ 1" term in the denominator:
an ≈ (9n^2)/(3n)
= 3n
Now, as n approaches infinity, the term 3n also approaches infinity. Therefore, the sequence diverges to positive infinity as n approaches infinity.
In conclusion, the sequence an = ((3n)2 )/(3n + 1) diverges.