Rewriting a binomial formula summation I am trying to obtain an explicit solution for $p$ from the following equation:
$p = 1 - \left[ \sum_{k=0}^{n} \binom{n}{k} p^k (1-p)^{n-k} \frac{1}{k+1} \right]^{\frac{1}{{n}}}$.
Obviously, I know that summing a binomial pdf we have
$\sum_{k=0}^{n} \binom{n}{k} p^k (1-p)^{n-k} = 1$
Perhaps there is a neat way to rewrite 
$\sum_{k=0}^{n} \binom{n}{k} p^k (1-p)^{n-k} \frac{1}{k+1}$
so I get an explicit solution for $p$.
 A: $$
\eqalign{
  & \sum\limits_{\left( {0\, \le \,} \right)\,k\,\left( { \le \,n} \right)} {{1 \over {k + 1}}\left( \matrix{
  n \cr 
  k \cr}  \right)p^{\,k} q^{\,n - k} }  = \sum\limits_{0\, \le \,k\,\left( { \le \,n} \right)} {{1 \over {n + 1}}\left( \matrix{
  n + 1 \cr 
  k + 1 \cr}  \right)p^{\,k} q^{\,n - k} }  =   \cr 
  &  = {1 \over {\left( {n + 1} \right)p}}\sum\limits_{0\, \le \,k\,\left( { \le \,n} \right)} {\left( \matrix{
  n + 1 \cr 
  k + 1 \cr}  \right)p^{\,k + 1} q^{\,\left( {n + 1} \right) - \left( {k + 1} \right)} }  =   \cr 
  &  = {1 \over {\left( {n + 1} \right)p}}\sum\limits_{1\, \le \,j\,\left( { \le \,n + 1} \right)} {\left( \matrix{
  n + 1 \cr 
  j \cr}  \right)p^{\,j} q^{\,\left( {n + 1} \right) - j} }  =   \cr 
  &  = {1 \over {\left( {n + 1} \right)p}}\left( {\sum\limits_{0\, \le \,j\,\left( { \le \,n + 1} \right)} {\left( \matrix{
  n + 1 \cr 
  j \cr}  \right)p^{\,j} q^{\,\left( {n + 1} \right) - j} }  - q^{\,\left( {n + 1} \right)} } \right) =   \cr 
  &  = {{1 - q^{\,n + 1} } \over {\left( {n + 1} \right)p}} = {1 \over {\left( {n + 1} \right)}}{{1 - q^{\,n + 1} } \over {1 - q}} \cr} 
$$
