# Thread: Why is the domain of arctan power series |x|<=1?

1. ## Why is the domain of arctan power series |x|<=1?

Why is the domain of arctan power series |x|<=1?

i understand why a power series has the domain |x|<1, but why is the power series of arctan |x|<=1?
thanks!

also, why are domains of the taylor expansions of coshx and sinhx: for all x?

2. Originally Posted by linyen416
Why is the domain of arctan power series |x|<=1?

i understand why a power series has the domain |x|<1, but why is the power series of arctan |x|<=1?
thanks!

also, why are domains of the taylor expansions of coshx and sinhx: for all x?
All power series can be viewed as an infinite series for a fixed x. Thus it makes sense to ask if the infinite sum we have converges or not. To check if an infinite series converges, we have many tests.
So do you know series tests? The ratio test, the root test etc? If you do, then you can figure out the domains.

3. Originally Posted by linyen416
Why is the domain of arctan power series |x|<=1?

i understand why a power series has the domain |x|<1, but why is the power series of arctan |x|<=1?
thanks!

also, why are domains of the taylor expansions of coshx and sinhx: for all x?
$\displaystyle \arctan x= x - \frac{x^3}{3} + \frac{x^5}{5} - ... + \frac{(-1)^n x^{2n+1}}{2n+1}$

$\displaystyle a_n = \frac{(-1)^n x^{2n+1}}{2n+1}$

Consider: $\displaystyle \mathop {\lim }\limits_{n \to \infty } \left|\frac{a_{n+1}}{a_n}\right|$

(Ratio Test)

4. Hi,
Originally Posted by Air
$\displaystyle a_n = \frac{(-1)^n x^{2n+1}}{2n+1}$

Consider: $\displaystyle \mathop {\lim }\limits_{n \to \infty } \left|\frac{a_{n+1}}{a_n}\right|$

(Ratio Test)
For $\displaystyle x=1$ you can't conclude using the ratio test. However, using the alternating series test, one can show that this series converges for any $\displaystyle x$ such that $\displaystyle |x|\leq 1$.