Answer:
The Taylor series for
, the first three non-zero terms are
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)
(2)
Substituting (2) in (1) we have:

Bring the factor
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
- If
the the series converges for all x - If
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

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