$ 1 + \frac 13 . \frac 14 + \frac 15 . \frac 1{4^2} + \frac 17 . \frac 1{4^3} + ...$
I evaluated the expression for the first few terms and I find that this number will probably tend to $ \log 3 $. I'd like to know why, or how I can prove that it does indeed tend to $ \log 3 $. More importantly, I'd like to know the relationship between this particular type of a series and the natural logarithm of numbers.
Why does the natural logarithm of a number show up in such a series?
Your series is: $$f(x) = \sum_{n=0}^\infty\dfrac{x^n}{2n+1}$$ with $x = \dfrac{1}{4}.$ The standard way to go about this kind of problem is to observe that $f(x)$ only converges uniformly for $|x|\leq r<1$ for any such $r$ and then manipulate the series into something we know already, usually involving taking derivatives.
If you want the challenge, then let me give you a hint: For $0<x<1$, consider $xf(x^2)$ and then take its derivative to get something you can easily calculate.