Q. 1 - Calculus

Question

If a sequence $a_1,a_2,a_3,\dots ,a_k,\dots$ approaches a real-number limit as $k\to \infty$, then we say that the sequence $\left\{a_k\right\}$converges. If the terms of the sequence do not get arbitrarily close to some real number, then we say that the sequence diverges. Write out enough terms of each sequence to make an educated guess as to whether it converges or diverges.

Part (a): $\left\{{\left(\frac{1}{4}\right)}^k\right\}$

Part (b): $\left\{{\left(\frac{5}{4}\right)}^k\right\}$

Part (c): $\left\{\frac{k}{k+2}\right\}$

Part (d): $\left\{\frac{k+1}{k}\right\}$

Step-by-Step Solution

Answer

Part (a): The function is convergent.

Part (b): The function is divergent. 

Part (c): The function is convergent. 

Part (d): The function is convergent. 

Part (a) Step 1. Given information.

Consider the given question,

$\left\{{\left(\frac{1}{4}\right)}^k\right\}$

Part (a) Step 2. Determine whether the function is convergent or divergent.

Consider the question,

$$\begin{gathered} f\left(k\right)={\left(\frac{1}{4}\right)}^k \\ \underset{k\to \infty }{\lim }{\left(\frac{1}{4}\right)}^k=0 \left(\because \underset{x\to \infty }{\lim } a^x=0 if \left|a\right|<1\right) \end{gathered}$$

Hence, the function is convergent.

Part (b) Step 1. Given information.

Consider the given question,

$\left\{{\left(\frac{5}{4}\right)}^k\right\}$

Part (b) Step 2. Determine whether the function is convergent or divergent.

Consider the question,

$$\begin{gathered} f\left(k\right)={\left(\frac{5}{4}\right)}^k \\ \underset{k\to \infty }{\lim }{\left(\frac{5}{4}\right)}^k=\infty \left(\because \underset{x\to \infty }{\lim } a^x=\infty if \left|a\right|>1\right) \end{gathered}$$

Hence, the function is divergent.

Part (c) Step 1. Given information.

Consider the given question,

$\left\{\frac{k}{k+2}\right\}$

Part (c) Step 2. Determine whether the function is convergent or divergent.

Consider the question,

$$\begin{gathered} f\left(k\right)=\frac{k}{k+2} \\ \underset{k\to \infty }{\lim }\left(\frac{k}{k+2}\right)=\underset{k\to \infty }{\lim }\left(\frac{1}{1+\frac{2}{k}}\right) \\ \underset{k\to \infty }{\lim }\left(\frac{k}{k+2}\right)=1 \end{gathered}$$

Hence, the function is convergent.

Part (d) Step 1. Given information.

Consider the given question,

$\left\{\frac{k+1}{k}\right\}$

Part (d) Step 2. Determine whether the function is convergent or divergent.

Consider the question,

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

Hence, the function is convergent.