Question #345755

If the power series {summation} an xn converges uniformly in ] α ,β [ then so does {summation} an (-x)n . true or false ? Justify


1
Expert's answer
2022-05-31T12:48:36-0400

ANSWER. The statement is true in the following cases :

1) the radius of convergence of the series n=0anxn\sum_{n=0}^{\infty}a_{n}x^{n} is \infty

2) the convergence set of the series is the segment [R,R][-R,R]

3) Points α,β\alpha, \beta are not end points of the set ]R,R[]-R,R[ (the convergence set of the series is the ]R,R]]-R,R] or [R,R[,or]R,R[[-R,R[, or ]-R,R[ )

If at least point one of the points α,β\alpha, \beta is the end point of the set indicated in 3) , then the statement is not true.

EXPLANATION

Note , that anxn=an(x)n|a_{n}x^{n}|=|a_{n}(-x)^{n}| . Therefore, if one of the series

n=0anxn\sum_{n=0}^{\infty}a_{n}x^{n} (1)

n=0an(x)n\sum_{n=0}^{\infty}a_{n}(-x)^{n} (2)

in a point cc absolutely converges , then the other converges absolutely. Since the power series in the interval of converges ]R,R[]-R,R[ converge absolutely , then the series (1) and (2) have the same radius of converges. Since the power series converges uniformly on any segment [α,β]]R,R[[\alpha, \beta]\subset ]-R,R[ , then both series (1), (2) converge uniformly on ]α,β[[α,β]]\alpha,\beta[\subset[\alpha, \beta] . This proves 1),2),3).

Let series (1) diverge at one of the point R,R-R,R (or both)

Suppose , that α=R\alpha=-R and both series (1), (2) converge uniformly on ]R,β[]-R,\beta[ . There are limits at the point (R)(-R) for all nNn\in \N

limx(R)anxn=an(R)n,limx(R)an(x)n=an(R)n.\lim_{x\rightarrow(-R)^{-}} {a_{n}}x^{n}=a_{n}(-R)^{n}\, \, , \, \lim_{x\rightarrow(-R)^{-}} {a_{n}}(-x) ^{n}=a_{n}( R)^{n}\, . By the theorem on the

limit of the sum of the power series we have the equalities

  limx(R)n=0anxn=n=0an(R)nlimx(R)n=0an(x)n=n=0anRn\lim_{x\rightarrow (-R)^{-}}\sum_{n=0}^{\infty} a_{n}x^{n}=\sum_{n=0}^{\infty} a_{n}(-R )^{n} \\\lim_{x\rightarrow (-R)^{-}}\sum_{n=0}^{\infty} a_{n}(-x) ^{n}=\sum_{n=0}^{\infty} a_{n}R^{n}

and the series on the right converge.Therefore, the series (1) converges on the [R,R][-R,R] in contrast to the assumption.


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