# Convergent or Divergent

1. Jul 3, 2014

### bigu01

1. The problem statement, all variables and given/known data
Which of the series, diverge or converge ∑ 5^n/(4^n +3 )

2. Relevant equations

3. The attempt at a solution Taking the limit as n→∞ we have (5^n ln 5)/ (4^n ln 4) , my question is here how does it become like this, which part am I missing here?

2. Jul 3, 2014

### bigu01

Oh I see, they have used L'hopital rule since we got infinity over infinity

3. Jul 3, 2014

### pasmith

A necessary condition for $\sum a_n$ to converge is $\lim_{n \to \infty} a_n = 0$ (it is not a sufficient condition; the series $\sum n^{-1}$ diverges). Here $$\lim_{n \to \infty} \frac{5^n}{4^n + 3}$$ is calculated using L'hopital's rule. Since the limit is not zero the sum does not converge.

4. Jul 3, 2014

### Zondrina

If $\Sigma a_n$ converges, then $\lim(a_n) = 0$.

If $\lim(a_n) ≠ 0$, then $\Sigma a_n$ diverges.

Alternatively, you could apply the comparison test + the geometric test with $|r| ≥ 1$.