Q. 31 - Calculus

Question

In Exercises 31–36 provide the first five terms of the given sequence. Unless specified, assume that the first term has index 1. 

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

Step-by-Step Solution

Answer

The first five terms are $2,0,\frac{2}{3},0,\frac{2}{5}$.

Step 1. Given information.

The given sequence is $\left\{\frac{1-(-1{)}^k}{k}\right\}$.

Step 2. First term.

The generals sequence will be,

$$\begin{gathered} a_k=\frac{1-(-1{)}^k}{k} \\ k=1, \\ {a}_1=\frac{1-(-1)}{k}=2 \end{gathered}$$

Step 3. Remaining terms.

Now put,

$$\begin{gathered} k=2, \\ {a}_2=\frac{1-(-1{)}^2}{k}=0 \\ k=3, \\ {a}_3=\frac{1-(-1{)}^3}{3}=\frac{2}{3} \\ k=4, \\ {a}_4=\frac{1-(-1{)}^4}{4}=0 \\ k=5, \\ {a}_5=\frac{1-(-1{)}^5}{5}=\frac{2}{5} \end{gathered}$$