## 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

Zbl 0664.40001
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