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Convergence-preserving functions: An alternative discussion. (English) Zbl 0783.40002

In an earlier article of Gerald Wildenberg [Am. Math. Mon. 95, No. 6, 542-544 (1988; Zbl 0664.40001)] it was shown that a function \(f\) on the reals has the property that \(\sum a_ n\) convergent implies \(\sum f(a_ n)\) convergent if and only if \(f(x)\) is a constant multiple of \(x\) in some neighborhood of the origin. The present paper gives a somewhat shorter proof which does not require considering the differentiability of such functions, and presents an example where \(\sum a_ n\) converges but \(\sum \sin a_ n\) does not.

MSC:

40A05 Convergence and divergence of series and sequences

Citations:

Zbl 0664.40001
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