Question

Test for convergence:$\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{2}}^{\mathbf{\infty }}\frac{\mathbf{1}}{\mathbf{n}\mathbf{l}\mathbf{n}{\mathbf{\left(}\mathbf{n}\mathbf{\right)}}^{\mathbf{3}}}$

Short answer

The series $\sum _{n=2}^{\infty }\frac{1}{n1n{\left(n\right)}^3}$diverges.

Step-by-step solution

Concept used to show that the series diverges

Integral test for convergence condition is used for given series which is expressed as follows:

For convergence:

If the function continues, decreasing and positive function then,

If ${\mathbf{\int }}_{\mathbf{k}}^{\mathbf{\infty }}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathit{d}\mathit{x}$ is convergent so is$\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{k}}^{\mathbf{\infty }}{\mathit{a}}_{\mathbf{n}}$ .

If ${\mathbf{\int }}_{\mathbf{k}}^{\mathbf{\infty }}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathit{d}\mathit{x}$is divergent so is$\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{k}}^{\mathbf{\infty }}{\mathit{a}}_{\mathbf{n}}$ .

Calculation to show that the series∑n=2∞1n1n(n)3 diverges

Let the function $f\left(x\right)$as$\frac{1}{x\ln \left(x^3\right)}$. For$x>0$ the function is positive and continues.

Now, integrate and apply the limit to the function as follows:

${\int }_2^{\infty }f\left(x\right)dx=li{m}_{t\to \infty }{\int }_2^t\frac{1}{x\ln \left(x^3\right)}dx$

${\int }_2^{\infty }f\left(x\right)dx=li{m}_{t\to \infty }{\int }_2^t\frac{1}{3x\ln \left(x\right)}dx$

$\ln \left(x\right)=p$

Now substitute the value in the above equation as follows:

$$\begin{gathered} {\int }_2^{\infty }f\left(x\right)dx=li{m}_{t\to \infty }{\int }_2^t\frac{1}{3p}dp \\ =li{m}_{t\to \infty }{\left(\frac{1}{3}\ln p\right)}_2^t \\ =li{m}_{t\to \infty }{\left(\frac{1}{3}\ln \left(\ln \right(x\left)\right)\right)}_2^t \\ =li{m}_{t\to \infty }\frac{1}{3}\left(\ln \right(\ln t\left)\right)-\left(\ln \right(\ln \left(2\right)\left)\right) \\ =li{m}_{t\to \infty }\frac{1}{3}\left(\ln \right(\ln \infty \left)\right)-\left(\ln \right(\ln \left(2\right)\left)\right) \\ =\infty \\ \backslash \mathrm{endgathered} \end{gathered}$$

Thus, the series diverges.