Find the Taylor series for f(x) centered at the given value a
f(x) = sin x, a = π/2
Also prove that the series obtained represents sin x for all x.
Assuming you know the Taylor series of sin and cos, one can try to use a little trigonometry, namely the Addition Formula for sin, to find the required Taylor series. And since I have written it up earlier, I am not going to repeat myself here.
Another way of figuring out the required Taylor series is, as I also wrote, to determine a general formula for the coefficent of in that series. If I am not mistaken, you will find the same value for both ways. - You are free choose the approach that you like better.
Is a regular formula (i would learn it!!) and is similar to completing the square. When we complete the square we don't change the value of the equation, we simply manipulate the information to get a desired result.
To factor this we do
If you expand that out we have
Same principle applies
And you can replace by as well, so all odd terms of the Taylor series will be 0.
Overall, with these simplifications you get:
Which is a roundabout way of arriving at a solution that you probably knew all along, namely that .