# Tagged Questions

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 ...
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, ...
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 ...
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 ...
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 ...
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 ...
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 ...
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