Radius and Interval of Convergence
Possible Behavior of
(1) The series converges at x = c and diverges elsewhere.
such that the series diverges (2) There is a positive number
but converges absolutely for for
The series may or may not converge at either of the endpoints
and
(3) The series converges absolutely for every x.
In case 2, the set of points at which the series converges is a called finite interval, called the Interval of Converges and
the Radius of Convergence of the series.
Radius of Convergence:
Given a power series,
so that the power series will There will exist a number
. This number and diverge for converge for
is called the radius of convergence.
Interval of Convergence:
The set of values of x for which the series is convergent is called the Interval of Convergence of the series .
Remark:
To find the Interval of Convergence or the Radius of Convergence we always apply the Ratio Test with absolute value .
i.e., Determining the radius of convergence of convergence for most power series is usually quite simple if we use the ratio test which is the following:
Given a power series compute,
, then
The series converges.
The series diverges.
The series may converge or diverge.
Example:
Determine the radius of convergence for the following power series:
.
Solution:
So, in this case we have
all we do is replace all the Remember that to compute
with n+1 n s in
Using the Ratio Test with absolute value then gives,
Now we know that the series will converge if,
and the series will diverge if,
In other words, the radius of the convergence for this series is
.
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