Recently I formulated the following curious conjecture based on my computation.

Conjecture. For all $|x|>1$, we have the identity $$\sum_{k=0}^\infty\frac{\sum_{j=0}^{2k}\binom{2k+1}{2j}(1-x)^jx^{k-j}}{(2k+1)(2x-1)^{2k+1}}=\frac1{2}\sum_{k=0}^\infty\frac{(-1)^k}{(2k+1)x^{k+1}}.\tag{1}$$

QUESTION. Is the conjecture true? Can one provide a proof of $(1)$?

I don't think the problem is very difficult. Your comments are welcome!

Recently I formulated the following curious conjecture based on my computation.

Conjecture. For all $|x|>1$, we have the identity $$\sum_{k=0}^\infty\frac{\sum_{j=0}^{k}\binom{2k+1}{2j}(1-x)^jx^{k-j}}{(2k+1)(2x-1)^{2k+1}}=\frac1{2}\sum_{k=0}^\infty\frac{(-1)^k}{(2k+1)x^{k+1}}.\tag{1}$$

QUESTION. Is the conjecture true? Can one provide a proof of $(1)$?

I don't think the problem is very difficult. Your comments are welcome!

Recently I formulated the following curious conjecture based on my computation.

Conjecture. For all $|x|>1$, we have the identity $$\sum_{k=0}^\infty\frac{\sum_{j=0}^{2k}\binom{4k+1}{2j}(1-x)^jx^{2k-j}}{(2k+1)(2x-1)^{4k+1}}=\frac1{2}\sum_{k=0}^\infty\frac{(-1)^k}{(2k+1)x^{k+1}}.\tag{1}$$

QUESTION. Is the conjecture true? Can one provide a proof of $(1)$?

I don't think the problem is very difficult. Your comments are welcome!

Recently I formulated the following curious conjecture based on my computation.

Conjecture. For all $|x|>1$, we have the identity $$\sum_{k=0}^\infty\frac{\sum_{j=0}^{2k}\binom{2k+1}{2j}(1-x)^jx^{k-j}}{(2k+1)(2x-1)^{2k+1}}=\frac1{2}\sum_{k=0}^\infty\frac{(-1)^k}{(2k+1)x^{k+1}}.\tag{1}$$

QUESTION. Is the conjecture true? Can one provide a proof of $(1)$?

I don't think the problem is very difficult. Your comments are welcome!