How far is \({\left( 1 + \frac{a}{n}\right) ^n} \) from \( e^a \) when a is an element of a Banach algebra?

  • Vito LampretEmail author
  • Philip G. Spain


We present effective upper and lower bounds for the distance from \(\displaystyle \left( 1 + \frac{a}{n}\right) ^n \) to \(\displaystyle e^a \) for an element a of a complex unital Banach algebra and positive integer n. Specifically
$$\begin{aligned} \frac{1}{2n} \sup \left\{ \left| \mathfrak {R}(z^2) \right| e^{\mathfrak {R}(z)} : z \in \sigma (a) \right\} \lesssim _{(2)} \left\| e^a - \left( 1 + \frac{a}{n}\right) ^n \right\| \le \frac{ \left\| a \right\| ^2}{2n} \ e^{ \left\| a \right\| }, \end{aligned}$$
where \(\sigma (a)\) is the spectrum of a. The symbol \(\lesssim _{(p)}\) means “less than or equal to, up to a term of order \(n^{-p}\)”as discussed below. Following some technical preliminaries (Sect. 1) we treat the real case (Sect.  2), extend to the complex case (Sect.  3), and then generalise to the case of a norm-unital Banach algebra (Sect.  4).


Asymptotic approximation Banach algebra Distance Estimate Exponential function Hermitian element Inequalities Norm Spectrum 

Mathematics Subject Classification

Primary: 40A25 41A99 46H99 Secondary: 26D99 47A50 65D20 



The authors are glad to take this opportunity to thank the referee for valuable comments and remarks which have enhanced the quality of the paper.


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    Lampret, V.: Approximating the powers with large exponents and bases close to unit, and the associated sequence of nested limits. Int. J. Contemp. Math. Sci. 6, 2135–2145 (2011)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.University of LjubljanaLjubljanaSlovenia
  2. 2.School of Mathematics and StatisticsUniversity of GlasgowGlasgowUK

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