Cheyanne Leigh

2021-03-12

Determine whether the series $\sum {a}_{n}$ an converges or diverges: Use the Alternating Series Test.

$\sum _{n=2}^{\mathrm{\infty}}(-1{)}^{n}\frac{n}{\mathrm{ln}(n)}$

Obiajulu

Skilled2021-03-13Added 98 answers

Consider the series $\sum _{n=2}^{\mathrm{\infty}}(-1{)}^{n}\frac{n}{\mathrm{ln}(n)}$

Take${a}_{n}=(-1{)}^{n}\frac{n}{\mathrm{ln}(n)}$ and ${b}_{n}=\frac{n}{\mathrm{ln}(n)}$

The Alternating series test is stated below:

Suppose the series$\sum {a}_{n}$ and either ${a}_{n}=(-1{)}^{n}{b}_{n}$ or ${a}_{n}=(-1{)}^{n+1}{b}_{n}$ where ${b}_{n}\ge 0$ for all n. Then if,

1.$\underset{n\to \mathrm{\infty}}{lim}{b}_{n}=0$

2.$\left\{{b}_{n}\right\}$ is a decreasing sequence

The series is convergent.

Check the first condition for series convergent.

$\underset{n\to \mathrm{\infty}}{lim}{b}_{n}=\underset{n\to \mathrm{\infty}}{lim}(\frac{n}{\mathrm{ln}(n)})$

$=\frac{\mathrm{\infty}}{\mathrm{\infty}}$

The value of the limit is in the indeterminate form.

Apply LHopitals rule to find the limit as follows.

$\underset{n\to \mathrm{\infty}}{lim}{b}_{n}=lim(\frac{\frac{d}{dn}(n)}{\frac{d}{dn}(\mathrm{ln}(n))})$

$=\underset{n\to \mathrm{\infty}}{lim}(\frac{1}{\frac{1}{n}})$

$=\underset{n\to \mathrm{\infty}}{lim}(n)$

$=\mathrm{\infty}$

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$\underset{n\to \mathrm{\infty}}{lim}{a}_{n}\ne 0$ then the series $\sum {a}_{n}$ will diverge".

Since$\underset{n\to \mathrm{\infty}}{lim}{b}_{n}\ne 0$ , the limit of the function ${a}_{n}=(-1{)}^{n}\frac{n}{\mathrm{ln}(n)}$ also not equal to zero as x goes to zero. That is, $\underset{n\to \mathrm{\infty}}{lim}{a}_{n}\ne 0$ .

By divergence test, it is concluded that the alternating series$\sum _{n=2}^{\mathrm{\infty}}(-1{)}^{n}\frac{n}{\mathrm{ln}(n)}$ diverges.

Take

The Alternating series test is stated below:

Suppose the series

1.

2.

The series is convergent.

Check the first condition for series convergent.

The value of the limit is in the indeterminate form.

Apply LHopitals rule to find the limit as follows.

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

Since

By divergence test, it is concluded that the alternating series

Jeffrey Jordon

Expert2021-12-27Added 2607 answers

Answer is given below (on video)

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