Part (a): Evaluate the Improper Integral

We are tasked to evaluate the improper integral:

$\int_{2}^\infty \frac{1}{x(\ln(x))^7} \, dx$

Steps:

1. Substitute: Let $u = \ln(x)$, so $du = \frac{1}{x} dx$. The limits of integration transform as follows:

   • When $x = 2$, $u = \ln(2)$,
   • When $x \to \infty$, $u \to \infty$.

   The integral becomes: $\int_{2}^\infty \frac{1}{x(\ln(x))^7} \, dx = \int_{\ln(2)}^\infty \frac{1}{u^7} \, du$

2. Integrate: The integral of $\frac{1}{u^7}$ is: $\int \frac{1}{u^7} \, du = \frac{-1}{6u^6}$

   Apply the limits: $\int_{\ln(2)}^\infty \frac{1}{u^7} \, du = \left[ \frac{-1}{6u^6} \right]_{\ln(2)}^\infty$

   At $u = \infty$, $\frac{-1}{6u^6} \to 0$. At $u = \ln(2)$, substitute $u = \ln(2)$: $\int_{\ln(2)}^\infty \frac{1}{u^7} \, du = 0 - \frac{-1}{6 (\ln(2))^6} = \frac{1}{6 (\ln(2))^6}$

Thus, the value of the improper integral is: $\boxed{\frac{1}{6 (\ln(2))^6}}$

Part (b): Convergence of the Series

The integral test states that if the improper integral converges, the corresponding series also converges. Since the improper integral:

$\int_{2}^\infty \frac{1}{x (\ln(x))^7} \, dx$

converges, the series:

$\sum_{n=2}^\infty \frac{1}{n (\ln(n))^7}$

also converges. Therefore, the answer is:

$\boxed{\text{the series converges}}$

Part (c): Approximation Using 6 Terms

To approximate $\sum_{n=2}^\infty \frac{1}{n (\ln(n))^7}$ using the first 6 terms, compute:

$\text{Terms: } \frac{1}{2 (\ln(2))^7}, \frac{1}{3 (\ln(3))^7}, \frac{1}{4 (\ln(4))^7}, \frac{1}{5 (\ln(5))^7}, \frac{1}{6 (\ln(6))^7}, \frac{1}{7 (\ln(7))^7}$

I'll compute these values.import numpy as np

Define the terms for the series approximation

n_values = np.arange(2, 8) # n = 2 to n = 7 terms = 1 / (n_values * (np.log(n_values))**7)

Compute the sum of the first 6 terms

approximation = np.sum(terms)

Display the individual terms and their sum

terms, approximation(array([6.50416840e+00, 1.72570905e-01, 2.54069078e-02, 7.15007180e-03, 2.81119383e-03, 1.35220481e-03]), 6.713459682493644)(array([6.50416840e+00, 1.72570905e-01, 2.54069078e-02, 7.15007180e-03, 2.81119383e-03, 1.35220481e-03]), 6.713459682493644)success### Results for Part (c): The individual terms are: