Determine whether the series sum a_n an converges or diverges: Use the Alternating Series Test. sum_{n=2}^infty(-1)^nfrac{n}{ln(n)}

Question
Series
Determine whether the series $$\sum a_n$$ an converges or diverges: Use the Alternating Series Test.
$$\sum_{n=2}^\infty(-1)^n\frac{n}{\ln(n)}$$

2021-03-13
Consider the series $$\sum_{n=2}^\infty(-1)^n\frac{n}{\ln(n)}$$
Take $$a_n=(-1)^n\frac{n}{\ln(n)}$$ and $$b_n=\frac{n}{\ln(n)}$$
The Alternating series test is stated below:
Suppose the series $$\sum a_n$$ and either $$a_n=(-1)^nb_n$$ or $$a_n=(-1)^{n+1}b_n$$ where $$b_n\geq0$$ for all n. Then if,
1.$$\lim_{n\to\infty}b_n=0$$
2. $$\left\{b_n\right\}$$ is a decreasing sequence
The series is convergent.
Check the first condition for series convergent.
$$\lim_{n\to\infty}b_n=\lim_{n\to\infty}(\frac{n}{\ln(n)})$$
$$=\frac{\infty}{\infty}$$
The value of the limit is in the indeterminate form.
Apply L'Hopital's rule to find the limit as follows.
$$\lim_{n\to\infty}b_n=\lim(\frac{\frac{d}{dn}(n)}{\frac{d}{dn}(\ln(n))})$$
$$=\lim_{n\to\infty}(\frac{1}{\frac{1}{n}})$$
$$=\lim_{n\to\infty}(n)$$
$$=\infty$$
Observe that, the limit of the sequence goes to infinity as x goes to infinity. Thus, the series does not converges.
The divergence test states that, "If $$\lim_{n\to\infty}a_n\ne0$$ then the series $$\sum a_n$$ will diverge".
Since $$\lim_{n\to\infty}b_n\ne0$$, the limit of the function $$a_n=(-1)^n\frac{n}{\ln(n)}$$ also not equal to zero as x goes to zero. That is, $$\lim_{n\to\infty}a_n\ne0$$.
By divergence test, it is concluded that the alternating series $$\sum_{n=2}^\infty(-1)^n\frac{n}{\ln(n)}$$ diverges.

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