Answer:
The Taylor series for  , the first three non-zero terms are
, the first three non-zero terms are  and the interval of convergence is
 and the interval of convergence is 
Step-by-step explanation:
<u>These are the steps to find the Taylor series for the function</u> 
- Use the trigonometric identity:

    2. The Taylor series of 
 
 
Substituting y=6x we have:
 
 
    3. Find the Taylor series for 
 (1)
 (1)
 (2)
 (2)
Substituting (2) in (1) we have:

Bring the factor  inside the sum
 inside the sum


Extract the term for n=0 from the sum:

<u>To find the first three non-zero terms you need to replace n=3 into the sum</u>

<u>To find the interval on which the series converges you need to use the Ratio Test that says</u>
For the power series centered at x=a

suppose that  . Then
. Then
- If  the the series converges for all x the the series converges for all x
- If  then the series converges for all then the series converges for all 
- If R=0, the the series converges only for x=a
So we need to evaluate this limit:

Simplifying we have:

Next we need to evaluate the limit

-(n+1)(2n+1) is negative when n -> ∞. Therefore 
You can use this infinity property  when a>0 and n is even. So
 when a>0 and n is even. So

Because this limit is ∞ the radius of converge is ∞ and the interval of converge is  .
.