Answered: k + 6 k k=1

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter9: Sequences, Probability And Counting Theory

Section9.4: Series And Their Notations

Problem 50SE: The sum of terms 50k2 from k=x through 7 is 115. What is x?

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Question

Determine whether the following series converge or diverge using the properties.

Expert Solution

Step 1

Given:

The series, $\sum_{k = 1}^{\infty} {(\frac{k + 6}{k})}^{k}$ .

Step 2

To determine whether the series converge or diverge using the properties.

Let, $\sum_{k = 1}^{\infty} {(\frac{k + 6}{k})}^{k}$ .

Divergence Test:

If $\lim_{n \to \infty} b_{n} \neq 0$ , then the series $\sum_{n = 1}^{\infty} b_{n}$ diverges.

Here, $b_{n} = {(\frac{k + 6}{k})}^{k}$ .

$\begin{array}{rcl} \Rightarrow & \lim_{k \to \infty} {(\frac{k + 6}{k})}^{k} = \lim_{k \to \infty} {(\frac{k}{k} + \frac{6}{k})}^{k} \\ = & \lim_{k \to \infty} {(1 + \frac{6}{k})}^{k} \\ = & e^{6} - - - - - \{∵ \lim_{n \to \infty} {(1 + \frac{t}{n})}^{n} = e^{t}\} \end{array} \neq 0$

Since the given series is not equal to zero, implies that, the series the diverges.

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