Hey,

the following function iapproaches - if the limit exists - the intuitive logarithm to base , i.e. the intuitive Abel function of developed at 1:

The question is whether this is indeed the logarithm, i.e. if for , provided that the limit exists at all.

It has a certain similarity to Euler's false logarithm series (pointed out by Gottfried here) as it can indeed be proven that for natural numbers (even for in difference to Euler's series):

if we now utilize that for then we get

Hence

But is this true also for non-integer ? Do we have some rules like , or even ?

the following function iapproaches - if the limit exists - the intuitive logarithm to base , i.e. the intuitive Abel function of developed at 1:

The question is whether this is indeed the logarithm, i.e. if for , provided that the limit exists at all.

It has a certain similarity to Euler's false logarithm series (pointed out by Gottfried here) as it can indeed be proven that for natural numbers (even for in difference to Euler's series):

if we now utilize that for then we get

Hence

But is this true also for non-integer ? Do we have some rules like , or even ?