For natural $A$ define $$ f(A)=\sum_{n=1}^\infty \frac{1}{A^n}\left(\frac{1}{An+1}- \frac{1}{An+A-1}\right)$$

$f(A)$ is BBP (Bailey-Borwein-Plouffe) formula and allows digit extraction in base $A$.

For small values of $A$, Wolfram Alpha finds closed form in simple terms of 2F1 and also very long closed form in terms of algebraic numbers and logarithms.

The elementary closed for for $f(5)$ is about one screen long, here it is in plaintext:

```
-3/4 - log(1 - 1/5^(1/5))/5^(4/5) - ((-1)^(2/5) log(1 - e^(-(2 i π)/5)/5^(1/5)))/5^(4/5) + ((-1)^(3/5) (log(-1 + e^((2 i π)/5)/5^(1/5)) - i π))/5^(4/5) - (-1/5)^(4/5) log(1 - e^(-(4 i π)/5)/5^(1/5)) +
((-1)^(1/5) (log(-1 + e^((4 i π)/5)/5^(1/5)) - i π))/5^(4/5) + 1/4 ((4 log(1 - 1/5^(1/5)))/5^(1/5) - (4 (-1)^(3/5) log(1 - e^(-(2 i π)/5)/5^(1/5)))/5^(1/5) + (4 (-1)^(2/5) (log(-1 + e^((2 i π)/5)/5^(1/5)) - i π))/5^(1/5)
- 4 (-1/5)^(1/5) log(1 - e^(-(4 i π)/5)/5^(1/5)) + (4 (-1)^(4/5) (log(-1 + e^((4 i π)/5)/5^(1/5)) - i π))/5^(1/5))
```

The elementary closed form for $f(7)$ is several screens long.

Q1 What is the intuition for so long elementary closed form?

Q2 Is there simpler elementary closed form for $f(A)$ in terms of $A$?

`[link text](problematic URL)`

will let it survive parsing (at least if you percent-escape any`)`

in the problematic URL. $\endgroup$n%29%5E%28-1%29+-+%28-1+%2B+A+%2B+An%29%5E%28-1%29%29%2FA%5En%2C%7Bn%2C1%2CInfinity%7D%5D $\endgroup$