Series Convergence Test for ∑ 5^n/(4^n +3)

[bigu01]

Homework Statement

Which of the series, diverge or converge ∑ 5^n/(4^n +3 )

Homework Equations

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?

 

[bigu01]

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

 

[pasmith]

Homework Statement

Which of the series, diverge or converge ∑ 5^n/(4^n +3 )

Homework Equations

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?

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.

 

[STEMucator]

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 $|r| ≥ 1$.