Thread: converging interval for power series

1. converging interval for power series

is it possible to have a powerr series that converges with interval of 0 inclusive to inf?

2. Re: converging interval for power series

$e^x = \sum \limits_{n=0}^\infty \frac1{n!}x^n$ converges everywhere.

Of course, you can't include "infinity" in the interval of convergence.

3. Re: converging interval for power series

the Taylor series of any function bounded on that interval converges everywhere on it.

for example

$\cos(x) = \sum \limits_{k=0}^\infty~\dfrac{(-1)^n x^{2n}}{2n!}$

converges for all $x \in \mathbb{R}$

4. Re: converging interval for power series

is there a reason why we cant have infinity like [0,inf)?

5. Re: converging interval for power series

Originally Posted by lc99
is there a reason why we cant have infinity like [0,inf)?
You want it to only converge on that interval?

I'm not understanding what you are asking.

6. Re: converging interval for power series

Originally Posted by lc99
is there a reason why we cant have infinity like [0,inf)?
No, there isn't. In his first response Archie was concerned that you "inclusive" referred to the "inf" as well as to 0. Yes, you can include 0 in an interval, no you cannot include "infinity" because that is not a number in the sense being used here. You can have [0, inf). You cannot have [0, inf].

7. Re: converging interval for power series

for a power series we cannot have an interval of convergence of the form $\displaystyle [0,\infty )$

the reason is that if a power series converges at $\displaystyle x=a\neq 0$

then it also converges for all $\displaystyle x$ with $\displaystyle |x|<|a|$