Is [math]’displaystyle ‘sum’infty_’n=0”frac'(-1)’n'(2n+1)”3′[/math] an irrational number?

A fairly easy-to-prove convergence criterion for alternating series is:

If the absolute amounts of the series members form a monotonously falling zero sequence, then the series converges, and if the series is broken for a particular row member, the “error” (i.e. the absolute amount of the deviation from the limit value) does not exceed the Absolute amount of the next (not included) row member.

According to this criterion, the series is certainly convergent.But the limit value is not 0, as Hinnerk Kändler suspects with a somewhat naive consideration: If one breaks off after the 5th row member (n = 4), then the five-member finite sum to six valid decimal places after the comma has exactly the value 0.969419 and after the above Criterion is the error (in the sense described above) less than 0.0014, so the limit cannot be 0! In fact, the following remarkable formula applies:

[math_sum _-n=0-infty-frac-(-1-) –__n————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————-

This is the answer to the actual question: Since [math’pi[/math is transcendent, this also applies to the limit value of this series, so this is not only not rational (i.e. irrational), but not even zero digit of any polynomial with only rational coefficients, how high the degree of polynomial and its rational coefficients are chosen!

By the way, the above formula for the limit value of the series is only the special case k = 1 of the following beautiful formula, which is valid for all integers k [math-ge [/math 0):

[math_sum _-n=0-infty-frac-(-1-1-) ————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————– .cdot – 2-2k+2-cdot ————————————————————————————————-

Here, [math’s [math’ denotes the k-th Euler number: These are whole (!) Numbers that can be easily recursively calculated starting with [math_E___0-=1[/math; see Euler’s Numbers – Wikipedia.By the way, this formula for k = 0 provides the famous series of Leibniz:

[math-frac-pi-4-=1–frac-1-3-1–frac-1-5–frac-1-7-1–frac-1-9–cdots [/math

However, the fact that the limit value of these series can be so beautifully stated in a “closed form” should not lead to the assumption that this can be done for all similar, simply constructed rows: the following rows:

[math-sum _-n=0-infty-frac-1-_n-__—————math

with whole k > 0 are certainly convergent (can be proved with the integral comparison criterion, see integral criterion – Wikipedia), but one does not even know here for all k > 1 whether the limit value is rational or irrational!For k = 1, the limit value has been proven to be irrational, but it is not known whether it is even transcendent; see e.g. Roger Apéry – Wikipedia.

And a “closed expression” similar to the above with [math-pi [/math and, if necessary, addition of the Euler number [mathe [/math is also not known; this would not necessarily be a contradiction to the fact that it is not known whether the limit value is rational or irrational, because, for example, no one knows whether the numbers [mathe+’pi [/math or [math’s’s’s’ pi[/math’ are rational or irrational, even though both individual numbers are transcendent!

Leave a Reply