Determining the convergence or divergence of a sequence using direct comparison

In summary, the conversation discusses the convergence of $\sum_{n=2}^{\infty} \frac{(\ln n)^{12}}{n^{9/8}}$ by comparing it to the divergent series $\sum_{n=1}^{\infty} \frac{1}{n^{1/8}}$. The speaker suggests proving that $\frac{(\ln n)^{12}}{n^{9/8}} \geq \frac{1}{n^{1/8}}$ to show divergence. They also mention the concept that $\ln n$ increases slower than any positive power of $n$ for convergence purposes.
  • #1
tmt1
234
0
I have

$$\sum_{n = 2}^{\infty} \frac{{(\ln\left({n}\right)})^{12}}{n^{\frac{9}{8}}}$$

We can compare it to $ \frac{1}{{n}^{\frac{1}{8}}}$. $ \sum_{n = 1}^{\infty} \frac{1}{{n}^{\frac{1}{8}}}$ diverges because $p < 1$ in this case. So, if I can prove that $ \frac{{(\ln\left({n}\right)})^{12}}{n^{\frac{9}{8}}} \ge \frac{1}{{n}^{\frac{1}{8}}}$ then that would mean $\sum_{n = 2}^{\infty} \frac{{\ln\left({n}\right)}^{12}}{n^{\frac{9}{8}}}$ diverges. Or $ \frac{{(\ln\left({n}\right)})^{12}}{n^{\frac{9}{8}}} - \frac{1}{{n}^{\frac{1}{8}}} \ge 0 $.

How can I prove this?
 
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  • #2
tmt said:
I have

$$\sum_{n = 2}^{\infty} \frac{{(\ln\left({n}\right)})^{12}}{n^{\frac{9}{8}}}$$

We can compare it to $ \frac{1}{{n}^{\frac{1}{8}}}$. $ \sum_{n = 1}^{\infty} \frac{1}{{n}^{\frac{1}{8}}}$ diverges because $p < 1$ in this case. So, if I can prove that $ \frac{{(\ln\left({n}\right)})^{12}}{n^{\frac{9}{8}}} \ge \frac{1}{{n}^{\frac{1}{8}}}$ then that would mean $\sum_{n = 2}^{\infty} \frac{{\ln\left({n}\right)}^{12}}{n^{\frac{9}{8}}}$ diverges. Or $ \frac{{(\ln\left({n}\right)})^{12}}{n^{\frac{9}{8}}} - \frac{1}{{n}^{\frac{1}{8}}} \ge 0 $.

How can I prove this?
The guiding principle is that $\ln n$ increases more slowly than any (positive) power of $n$. So for convergence purposes $\sum\frac{(\ln n)^{12}}{n^{9/8}}$ behaves pretty much like $\sum\frac1{n^{9/8}}$. Can you take it from there?
 

FAQ: Determining the convergence or divergence of a sequence using direct comparison

What is direct comparison in determining the convergence or divergence of a sequence?

Direct comparison is a method used to determine the convergence or divergence of a sequence by comparing it to another known sequence with known convergence properties. This allows us to make conclusions about the convergence or divergence of the given sequence.

When is direct comparison applicable in determining the convergence or divergence of a sequence?

Direct comparison is applicable when the given sequence is positive and the known sequence has a known convergence or divergence property. It is also applicable when the given sequence can be expressed as a multiple or a ratio of the known sequence.

How do you use direct comparison to determine the convergence or divergence of a sequence?

To use direct comparison, you compare the given sequence to the known sequence term by term. If the known sequence converges and the given sequence is always less than or equal to the known sequence, then the given sequence also converges. If the known sequence diverges and the given sequence is always greater than or equal to the known sequence, then the given sequence also diverges.

Can direct comparison be used to prove absolute convergence?

Yes, direct comparison can be used to prove absolute convergence. If the given sequence is absolute convergent and can be expressed as a multiple or a ratio of the known sequence, then the given sequence is also absolutely convergent.

What are the limitations of direct comparison in determining the convergence or divergence of a sequence?

Direct comparison can only be used when the given sequence is positive and when there is a known sequence with known convergence properties to compare it to. It also may not always provide a definitive answer and further methods may need to be used to determine the convergence or divergence of a sequence.

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