Let (a n ) be a sequence of positive numbers. Prove that Conclude that whenever the ratio test determines convergence or divergence, the root test does, too. [Hint: To prove the inequality on the...

Let (a_n) be a sequence of positive numbers. Prove that

Conclude that whenever the ratio test determines convergence or divergence, the root test does, too. [Hint: To prove the inequality on the right, let α = lim sup a_n+1/a_n. If α = + ∞, the result is obvious. If α is finite, choose β > α . Also, there exists N ∈ N such that an + 1/a_n
<>

n ≥ N. That is, for n ≥ N we have

a_n
<>_n−1, a_n−1
<>_n−2,…….., a_N+1

Combine these n – N inequalities to obtain an
ⁿ
for a positive constant c. Argue from this that lim sup (a_n)^1/n
≤ β . Since this holds for each β > α, the desired inequality follows.]