A lawyer drives from her home, located 3 miles east and 5 miles north of the town courthouse, to her office, located 6 miles wes
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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 .