1

$$\sum_{n=1}^{\infty} \ln\left(\dfrac{n^4+3n^3}{n^4+1}\right)$$ using the characteristic comparison, to investigate the series for convergence.

Martin Sleziak
  • 53,687
Anastasia
  • 11
  • 1
    If you would use MathJax, I would be able to help you, but I cannot see your image. – gt6989b Nov 03 '16 at 16:17
  • why not? click on the link(( – Anastasia Nov 03 '16 at 16:48
  • 1
    Well, @Anastasia, go ahead and write down what you tried. As you might have noticed by the trend of you r previous questions, this site does not appreciate homework being dumped without effort. –  Nov 03 '16 at 17:26
  • @Anastasia because the site is restricted from my work's access – gt6989b Nov 03 '16 at 17:42

1 Answers1

1

In THIS ANSWER, I showed using only the limit definition of the exponential function along with Bernoulli's Inequality that the logarithm function satisfies the inequalities

$$\bbox[5px,border:2px solid #C0A000]{\frac{x-1}{x}\le \log(x)\le x-1} \tag 1$$

Using the left-hand side inequality in $(1)$, we can assert that

$$\log\left( \frac{n^4+3n^3}{n^4+1}\right)\ge \frac{3n^3-1}{n^4+3n^3}\ge \frac1{4n}$$

We conclude by comparison with the harmonic series that the series of interest diverges.

Community
  • 1
Mark Viola
  • 179,405