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Prove by induction that (1+1/2)!+1+1/4!+1/8!+…+1/2^n!=2^n-1!+2

I’m trying to prove that $$(1+1/2)!+1+1/4!+1/8!+…+1/2^n!=2^n-1!+2$$
But I’m stuck. Would someone please help me out?

A:

Note that we have $2^{n-1} – 1 = (2^{n-1} – 1) + 1$, so we can rewrite
$$
\sum_{k=1}^n \frac{1}{2^k} + (2^n – 1) = 2^n \left( 1 + \frac{1}{2^n} + \frac{1}{2^{2n}} + \cdots \frac{1}{2^{(n-1)n}} + \frac{1}{2^{(n+1)n}} \right).
$$
Now, what happens if we write down the most significant term from the above sum?

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