Q. 26 - Calculus

Question

In Exercises $21-30$ use one of the comparison tests to determine whether the series converges or diverges. Explain how the given series satisfies the hypotheses of the test you use.

$\sum _{k=1}^{\infty } \frac{1}{k+0.01}$.

Step-by-Step Solution

Answer

The series $\sum _{k=1}^{\infty } \frac{1}{k+0.01}$ is divergent.

Step 1 . Given information

$\sum _{k=1}^{\infty } \frac{1}{k+0.01}$.

Step 2 . The comparison test states that for ∑ k = 1 ∞   a k and ∑ k = 1 ∞   b k be two series with positive term then,

1. If $\underset{k\to \infty }{\lim } \frac{a_k}{b_k}=L$ where $L$ is any positive real number, then either both converge or both diverge.
2. If $\underset{k\to \infty }{\lim } \frac{a_k}{b_k}=0$ and $\sum _{k=1}^{\infty } b_k$ converges, then $\sum _{k=1}^{\infty } a_k$ converges.
3. If $\underset{k\to \infty }{\lim } \frac{a_k}{b_k}=\infty$ and $\sum _{k=1}^{\infty } b_k$ diverges, then $\sum _{k=1}^{\infty } a_k$ diverges.

Step 3 . The terms of the series ∑ k = 1 ∞   1 k + 0 . 01 are positive.

Find $\sum _{k=1}^{\infty } b_k$ for the given series.

$\sum _{k=1}^{\infty } b_k=\sum _{k=1}^{\infty } \frac{1}{k}$

Next find $\underset{k\to \infty }{\lim } \frac{a_k}{b_k}$ for the given series.

$$\begin{gathered} \underset{k\to \infty }{\lim } \frac{a_k}{b_k}=\underset{k\to \infty }{\lim } \frac{{\displaystyle \frac{1}{k+0.01}}}{{\displaystyle \frac{1}{k}}} \\ =\underset{k\to \infty }{\lim } \frac{k}{k+0.01} \\ =\underset{k\to \infty }{\lim } \frac{k}{k\left(1+{\displaystyle \frac{0.01}{k}}\right)} \\ =\underset{k\to \infty }{\lim } \frac{1}{\left(1+{\displaystyle \frac{0.01}{k}}\right)} \\ =1 \end{gathered}$$

Step 4 . From the obtained values,

The value of $\underset{k\to \infty }{\lim } \frac{a_k}{b_k}=1$ which is a finite non zero number.

The value of $\sum _{k=1}^{\infty } b_k=\sum _{k=1}^{\infty } \frac{1}{k}$ is divergent by $p-$series test.

Therefore, the series $\sum _{k=1}^{\infty } a_k$ is also divergent.

Hence, the given series is divergent.