Q. 33 - 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. 

$a_k=\frac{\cos \left(kx\right)}{x^k+k^2}$

Step-by-Step Solution

Answer

The first five terms are $\frac{\cos \left(x\right)}{x+1},\frac{\cos \left(2x\right)}{x^2+4},\frac{\cos \left(3x\right)}{x^3+9},\frac{\cos \left(4x\right)}{x^4+16},\frac{\cos \left(5x\right)}{x^5+25}$.

Step 1. Given information.

The given series is $a_k=\frac{\cos \left(kx\right)}{x^k+k^2}$.

Step 2. First term.

The general sequence is $a_k=\frac{\cos \left(kx\right)}{x^k+k^2}$.

Put $k=1,$

$$\begin{gathered} a_1=\frac{\cos \left(1x\right)}{x^1+1^2} \\ =\frac{\cos \left(x\right)}{x+1} \end{gathered}$$

Step 3. Remaining terms.

Now, put,

$$\begin{gathered} k=2, \\ {a}_2=\frac{\cos \left(2x\right)}{x^2+2^2} \\ k=3 \\ {a}_3=\frac{\cos \left(3x\right)}{x^2+3^2} \\ k=4 \\ {a}_4=\frac{\cos \left(4x\right)}{x^4+{16}^2} \\ k=5 \\ {a}_5=\frac{\cos \left(5x\right)}{x^4+5^2} \end{gathered}$$