Around the point " x " = 0, we find out that the radius of convergence of this series is \ scriptstyle \ infty meaning that this series converges for all complex numbers.
42.
Another problem of extrapolation is loosely related to the problem of analytic continuation, where ( typically ) a power series representation of a convergence to produce a power series with a larger radius of convergence.
43.
Hi there guys-I was wondering about determining the convergence of the series \ sum ^ { \ infty } z ^ { n ! }-particularly on the boundary of the radius of convergence.
44.
From above it is evident that all that is really needed is the radius of convergence of the MacLaurin series be greater than ? " T " ?, the operator norm of " T ".
45.
To prove ( i ) and ( v ), apply the ratio test and use formula ( 2 ) above to show that whenever " ? " is not a nonnegative integer, the radius of convergence is exactly 1.
46.
Conversely, for a given holonomic sequence c _ n, the function defined by the above sum is holonomic ( this is true in the sense of formal power series, even if the sum has a zero radius of convergence ).
47.
For a value such as " t " = 0.000000001 that is significantly smaller than this radius of convergence, the first-order approximation 19.9569 . . . is reasonably close to the root 19.9509 . ..
48.
Is a formal power series with rational coefficients " a " " n ", which has a non-zero radius of convergence in the complex plane, and within it represents an analytic function that is in fact an algebraic function.
49.
If the inner radius of convergence is positive, " f " may have infinitely many negative terms but still be regular at " c ", as in the example above, in which case it is represented by a " different"
50.
And one expects problems when | " t " | is larger than the radius of convergence of this power series, which is given by the smallest value of | " t " | such that the root ? " j " becomes multiple.
