I have been trying to find the value of $\sum_{n=1}^\infty\frac{1}{2^n-1}$, But I dont really know how so I've tried a few things and managed to express it (maybe unusefully) as $\sum_{n=1}^\infty\sum_{m=1}^\infty\frac{1}{2^{nm}}$
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It seems that there is no analytical formula for the sum. Or perhaps you need to use some special functions. – NN2 Nov 16 '22 at 09:54
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Check Wolfram alpha. It seems there is no "nice" closed form. https://www.wolframalpha.com/input?i=Sum%5B1%2F%282%5En+-1%29%2C%7Bn%2C1%2CInfinity%7D%5D – Zubzub Nov 16 '22 at 10:06