Can Logarithmic Functions Be Expressed as Infinite Series?

[MupptMath]

Hi,

I'm stuck on the following proof:

\(\displaystyle \log[3] = \frac 1{729} \sum_{k=0}^\infty \frac 1{729^k} \left[\frac{729}{6k+1}+\frac{81}{6k+2}+\frac{81}{6k+3}+\frac 9{6k+4}+\frac 9{6k+5}+\frac 1{6k+6}\right]\)

Manipulating and converting summands to integrals of the form $x^{-(6k+n)}$ over {x,0,3} seems to provide a nice reduction but does not converge in that interval and am uncertain how to proceed. Manipulating indices and summands to reduce to certain series doesn't quite work. Seems so close, but so far.

Thanks.

 

[jvicens]

Hi there,

Thank you for reaching out for help with this proof. After looking at the equation and trying out a few manipulations myself, I have a few suggestions that may help you make progress.

Firstly, when dealing with logarithms, it can be helpful to use the logarithm rules to manipulate the equation. In this case, you can use the rule $\log_a(bc) = \log_a(b) + \log_a(c)$ to split the logarithm into multiple terms. This may help simplify the equation and make it easier to work with.

Additionally, when dealing with infinite sums, it can be helpful to look for patterns or common terms that can be factored out. In this case, you may notice that the terms inside the square brackets have a common factor of 9. You can use this to simplify the equation and possibly find a way to convert it into a convergent series.

Lastly, I would suggest revisiting the integral approach you mentioned. Sometimes, integrating over a different interval or using a different variable can help with convergence issues.

I hope these suggestions are helpful and that you are able to make progress on your proof. Good luck!