Tagged Questions
0
votes
2answers
40 views
Taylor Series of $\sin 2x$ finding $f^{(n)} (a)$ where $a = 0$
ok so i get;
f (x) = sin 2x
f ' = 2cos 2x
f '' = -4sin 2x
f ''' = -8cos 2x
f '''' = 16sin 2x
f ''''' = 32cos 2x
f (0) = 0
f '(0) = 2
f ''(0) = 0
f '''(0) = -8
f ''''(0) = 0
f '''''(0) = 32
...
2
votes
2answers
165 views
Finding the 9th derivative of $\frac{\cos(5 x^2)-1}{x^3}$
How do you find the 9th derivative of $(\cos(5 x^2)-1)/x^3$ and evaluate at $x=0$ without differentiating it straightforwardly with the quotient rule? The teacher's hint is to use Maclaurin Series, ...
3
votes
2answers
82 views
Infinite (Taylor) Series
What is the general term for a Taylor series of $\sin(x)$ centered at $\pi/4$?
It should be $(-1)^{[??]} \times \sqrt{2}/2 \times \frac{(x-\pi/4)^n}{n!}$
What power is $(-1)$ supposed to be raised ...
3
votes
1answer
253 views
Approximating $\arctan x$ for large $|x|$
I would like to know if there is reasonably fast converging method for computing large arguments of arctan.
Until now I've came across Taylor series that converges only on interval $(-1,1)$ and for ...
4
votes
2answers
304 views
How do we know Taylor's Series works with complex numbers?
Euler famously used the Taylor's Series of $\exp$:
$$\exp (x)=\sum_{n=0}^\infty \frac{x^n}{n!}$$
and made the substitution $x=i\theta$ to find
$$\exp(i\theta) = \cos (\theta)+i\sin (\theta)$$
How ...
22
votes
2answers
416 views
Is the Maclaurin series expansion of $\sin x$ related to the inclusion-exclusion principle?
When I see the alternating signs in the infinite series expansion of $\sin x$, I'm reminded of the inclusion-exclusion principle. Could there be any way to visualize it in such a way?
Also, is there ...
1
vote
1answer
192 views
Taylor series expansion of $\sec(x +y^2)$
We have $f(x,y) = \sec(x+y^2)$
I want to find the first two non-zero terms of $f$ at $(0,0)$ starting by
Taking the first few terms of $\cos x$ centered at zero, $1 - \frac{x^2}{2!} $
Using this ...
