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

Article

## 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 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).

## 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

## References

1. 1.
Bonsall, F.F., Duncan, J.: Complete normed algebras. Springer, Berlin (1973)
2. 2.
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)