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

Smith, Arthur

Am. Math. Mon. 98, No. 9, 831-833 (1991).

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.

Reviewer: G.A.Heuer (Moorhead)

MSC:

40A05

Convergence and divergence of series and sequences

Keywords:

convergence; convergence-preserving functions; Cauchy’s functional equation

Citations:

Zbl 0664.40001