An archived instance of discourse for discussion in undergraduate Real and Numerical Analysis.

# Summing a (binary) expansion

NumericalAudrey

Note: This question has been edited to reflect the content we covered in class.

Problem number 2.2(b) asks us to express $101_{\dot 2}10\overline{011}$ as a fraction. How should we do this?

Ricky_Bobby

Great question Audrey!!!

Cheryl

I'm not certain if this is correct but here's what I came up with...
$$101.10\overline{011} = 2^2+2^0+\frac{1}{2^2} \Bigg(1+ \sum_{k=1}^{\infty} \frac{3}{2^{2k}} \Bigg)$$
$$=4+1+\frac{1}{2^2}\Bigg(1+\sum_{k=1}^{\infty}\frac{3}{2^{2k}}\Bigg)$$
$$=5+\frac{1}{2^2}\Bigg(1+\sum_{k=1}^{\infty}\frac{3}{2^{2k}}\Bigg)$$
Please correct this if wrong and show where the mistake is.

mark

@Cheryl Very close - definitely worth a like! I think there's one little goof, though. I'm not convinced that you've shifted the repeating portion the correct amount.

XCNate

Just a quick, small adjustment to the shift: $$101.10\overline{011} = 2^2 + 2^0 + \frac{2}{2^2}+ \frac{1}{2^2}\sum_{k=1}^{\infty}\frac{3}{2^{3k}}$$
$$= 5 + \frac{1}{2^2}\left(2 + \sum_{k=1}^{\infty}\frac{3}{2^{3k}}\right)$$