Question

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?

Answers

MathMate
1/(1-x) = Σ<sub>0</sub><sup>∞</sup>x<sup>k</sup>

write 1/(2-x) as (1/2)(1/(1-x/2))
then
1/(2-x)
= (1/2)(1/(1-x/2))
= (1/2)Σ<sub>0</sub><sup>∞</sup>x/2<sup>k</sup>
=Σ<sub>0</sub><sup>∞</sup>x<sup>k+1</sup>

as you have corrected determined.

MathMate
Sorry, the last two lines should read:

=(1/2)Σ<sub>0</sub><sup>∞</sup>(x/2)<sup>k</sup>
=Σ<sub>0</sub><sup>∞</sup>x<sup>k</sup>/2<sup>k+1</sup>
Sean
Thanks so much!

I can't read your symbols, but I can make out what you are doing and it results in the right answer.
MathMate
Sorry, I was accidentally on unicode.
You would be able to read the symbols with unicode encoding. If you are on Windows XP, you can use the menu to go
view/character encoding/utf-8
However, utf-8 may not be available automatically on all computers.

I will take more care with the encoding next time. Thanks.

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