If the power series {summation} an xn converges uniformly in ] α ,β [ then so does {summation} an (-x)n . true or false ? Justify
ANSWER. The statement is true in the following cases :
1) the radius of convergence of the series is
2) the convergence set of the series is the segment
3) Points are not end points of the set (the convergence set of the series is the or )
If at least point one of the points is the end point of the set indicated in 3) , then the statement is not true.
EXPLANATION
Note , that . Therefore, if one of the series
(1)
(2)
in a point absolutely converges , then the other converges absolutely. Since the power series in the interval of converges converge absolutely , then the series (1) and (2) have the same radius of converges. Since the power series converges uniformly on any segment , then both series (1), (2) converge uniformly on . This proves 1),2),3).
Let series (1) diverge at one of the point (or both)
Suppose , that and both series (1), (2) converge uniformly on . There are limits at the point for all
By the theorem on the
limit of the sum of the power series we have the equalities
and the series on the right converge.Therefore, the series (1) converges on the in contrast to the assumption.
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