Obtain the MacLaurin series for 1/(2-x) by making an appropriate substitution into the MacLaurin series for 1/(1-x).

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The MacLaurin series for 1/(1-x) = Σ x^k

I substitue (x-1) in for x, because 1/(2-x) = 1/(1-(x-1))

Making the same substitution in the MacLaurin series gives Σ (x-1)^k

If I manually calculate the MacLaurin series for 1/(2-x), I get Σ x^k/2^(k+1)

Those two don't match. What did I do wrong? Or does that substitution method not work?

------------

The MacLaurin series for 1/(1-x) = Σ x^k

I substitue (x-1) in for x, because 1/(2-x) = 1/(1-(x-1))

Making the same substitution in the MacLaurin series gives Σ (x-1)^k

If I manually calculate the MacLaurin series for 1/(2-x), I get Σ x^k/2^(k+1)

Those two don't match. What did I do wrong? Or does that substitution method not work?

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