Does the Series $\sum_{n=2}^{\infty} (-1)^n \frac{4}{5\ln{n}}$ Converge?

  • MHB
  • Thread starter karush
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In summary, "10.6.10 converge or diverge" refers to a series that may either approach a specific value (converge) or continue to increase without limit (diverge). To determine if a series converges or diverges, various tests such as the ratio test, the comparison test, or the integral test can be used. A convergent series approaches a finite value, while a divergent series either approaches infinity or does not have a well-defined limit. A series can only either converge or diverge, but different series can converge or diverge to the same value. If a series is conditionally convergent, it means that although the series itself converges, it does not converge absolutely.
  • #1
karush
Gold Member
MHB
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$\tiny{10.6.10}\\ $
$\textsf{ converge or diverge?}\\$
\begin{align*}\displaystyle
S_n&= \sum_{n=2}^{\infty} (-1)^n \frac{4}{5\ln{n}}\\
&\frac{4}{5} \sum_{n=2}^{\infty} (-1)^n
\frac{1}{\ln{n}}=
\end{align*}
?
??
?

$\textit{converges: alternating series test}$
 
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  • #2
karush said:
$\tiny{10.6.10}\\ $
$\textsf{ converge or diverge?}\\$
\begin{align*}\displaystyle
S_n&= \sum_{n=2}^{\infty} (-1)^n \frac{4}{5\ln{n}}\\
&\frac{4}{5} \sum_{n=2}^{\infty} (-1)^n
\frac{1}{\ln{n}}=
\end{align*}
?
??
?

$\textit{converges: alternating series test}$

Yes it converges as ln(n) is a decreasing function.
 

FAQ: Does the Series $\sum_{n=2}^{\infty} (-1)^n \frac{4}{5\ln{n}}$ Converge?

What does "10.6.10 converge or diverge" mean?

"10.6.10 converge or diverge" refers to the series 10.6 + 10 + 10 + ... where the terms are added indefinitely. The question is asking whether this series will approach a specific value (converge) or continue to increase without limit (diverge).

How do you determine if a series converges or diverges?

To determine if a series converges or diverges, you can use various tests such as the ratio test, the comparison test, or the integral test. These tests involve evaluating the limit of the series and comparing it to known values in order to determine if the series converges or diverges.

What is the difference between a convergent and divergent series?

A convergent series is one that approaches a finite value as more terms are added, while a divergent series is one that either approaches infinity or does not have a well-defined limit. In other words, a convergent series has a finite sum, while a divergent series does not.

Can a series both converge and diverge?

No, a series can only either converge or diverge. It cannot do both simultaneously. However, it is possible for different series to converge to the same value or diverge to the same limit.

What does it mean if a series is conditionally convergent?

If a series is conditionally convergent, it means that although the series itself converges, it does not converge absolutely. In other words, the series only converges if the terms are added in a specific order, but if the terms are rearranged, the series may diverge. This is in contrast to absolutely convergent series, where the series converges regardless of the order of the terms.

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