Exploring the Value and Representations of ee

The constant e, approximately 2.71828, is a fundamental number in mathematics. One of its interesting forms is ee. This article delves into the evaluation and representations of ee, exploring its value, inequalities, and various mathematical expressions.

Value of ee

The value of ee can be approximated as:

ee ≈ 2.718282.71828 ≈ 15.15426

For a more precise value, here is e raised to the power of e given to 800 decimal digits (verified with Mathematica):
15.15426224147926418976043027262991190552854853685613976914074640591483097373093443260845696835787346051158726885285229584108349266426657649118779479704154810461761622938836845482194326518823698067581131232299035461333833518596595421652507204871131694841248837028298101630940495747791991372453217285387321910680977914733658187699967694174778649038163390505612049776125348054466629607940201952987727518553087967728180527535931123975906005188808804151764154263227653969369419281681418048811050162285713125125736860841705024753725516254728475141045799649334649258377732997799526746207088566625779404589544900951646188503245155543276102551379333718085468414791771323547050692212614636013851810485295066335920575541400093728813275661177976041869730169672487165342920993670102150408829499597506617616719843

Continued Fraction Representation

The continued fraction representation of ee can be expressed in several forms. Here, we present a compact form:

ee ≈ (frac{sqrt{1867854217}45949}{5884})

This form provides a good approximation of the numerical value of ee to 12 decimal digits.

Inequalities and Basic Properties

Considering the basic inequality:

ex ≥ x 1 (for all real x, with equality if and only if x 0)

Applying this to x e, we have:

ee ≥ e 1

Other Representations of ee

There are several other mathematical expressions that represent ee. Here are a few notable ones:

1. Series Representation:

ee (left(frac{1}{sum_{k0}^{infty} frac{(-1)^k}{k!}}right)^{left(frac{1}{sum_{k0}^{infty} frac{(-1)^k}{k!}}right)})

ee (4^{left(sum_{k0}^{infty} frac{k-1}{2k-1!}right)} left(sum_{k0}^{infty} frac{k-1}{2k-1!}right)^{left(2 sum_{k0}^{infty} frac{k-1}{2k-1!}right)})

ee (left(sum_{k0}^{infty} frac{8k-4(8k^2-1)}{4k!}right)^{left(sum_{k0}^{infty} frac{3-4k^2}{2k-1!}right)})

2. Product Representation with Moebius Function:

ee (left(prod_{k1}^{infty} k^{-frac{mu(k)}{k}}right)^{left(prod_{k1}^{infty} k^{-frac{mu(k)}{k}}right)})

3. Generalized Hypergeometric Function Form:

ee $_0F_0 1^{_{0F_0 1}}$

4. Meijer G-Function Representation:

ee $G_{01}^{10}left(-1left(begin{array}{c} 0 end{array}right)right)^{G_{01}^{10}left(-1left(begin{array}{c} 0 end{array}right)right)}$

Conclusion

In summary, ee is a fascinating mathematical expression with rich representations. From basic inequalities to more complex series and functional forms, ee demonstrates the beauty and complexity inherent in mathematical constants. Understanding and exploring these representations enhances our appreciation of the underlying mathematical structures.