Determine whether the sequence converges or diverges. If it converges, find the limit.

Question

a_n=(6ⁿ)/(1+7ⁿ)

Answer 1

Notice that 6ⁿ/7ⁿ > 6ⁿ/(1+7ⁿ), and we can tell that 6ⁿ/7ⁿ converges to 0 because 7ⁿ in the denominator grows faster than 6ⁿ in the numerator. Thus, by the Direct Comparison Test, since the larger series converges, you can say that the smaller series, 6ⁿ/(1+7ⁿ), also converges:

lim_n->∞ 6ⁿ/(1+7ⁿ) = 0

Hope this helped!

Answer 2

To determine if the sequence converges or diverges, we need to take the limit of the sequence as n approaches infinity.

As n approaches infinity, (1+7ⁿ) ~ 7ⁿ. Then, we can easily compute the limit as n approaches infinity for (6/7)ⁿ, which is equal to 0. Therefore, the sequence converges to 0.