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

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## Abstract

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 where \(\sigma (a)\) is the spectrum of

*a*of a complex unital Banach algebra and positive integer*n*. Specifically$$\begin{aligned} \frac{1}{2n} \sup \left\{ \left| \mathfrak {R}(z^2) \right| e^{\mathfrak {R}(z)} : z \in \sigma (a) \right\} \lesssim _{(2)} \left\| e^a - \left( 1 + \frac{a}{n}\right) ^n \right\| \le \frac{ \left\| a \right\| ^2}{2n} \ e^{ \left\| a \right\| }, \end{aligned}$$

*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).## Keywords

Asymptotic approximation Banach algebra Distance Estimate Exponential function Hermitian element Inequalities Norm Spectrum## Mathematics Subject Classification

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

### Acknowledgements

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.

## References

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**6**, 2135–2145 (2011)MathSciNetzbMATHGoogle Scholar

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