A closed form of $\frac{1}{2k+1}+\frac{2k}{(2k+1)(2k-1)}+\ldots+\frac{2k(2k-2)\cdots 4}{(2k+1)(2k-1)\cdots 3}$

Question

Is there a closed form representation of the following sum?

$$\frac{1}{2k+1}+\frac{2k}{(2k+1)(2k-1)}+\frac{2k(2k-2)}{(2k+1)(2k-1)(2k-3)}+\cdots+\frac{2k(2k-2)\cdots 6\cdot 4}{(2k+1)!!}.$$

Here $k$ is a positive integer and $!!$ is the double factorial.

Answer 1

Let this sum be $S_k$; then $S_1=1/3$ and $(2k+1)S_k=1+2kS_{k-1}$ for $k>1$. The closed form is $$S_k=1-\frac{(2k)!!}{(2k+1)!!}$$ because it satisfies the same recurrence and initial condition (therefore it holds by induction on $k$).

Simply put, if you stop the summation at $\frac{2k(2k-2)\cdots 2}{(2k+1)(2k-1)\cdots 1}$ instead, the sum becomes equal to $1$.