Question

Write the binary number
\[ 11,101\overline{110}\]
as decimal fraction and explain it with help of geometric series.

Answer 1

your comma means a binary point right ?
(like decimal point)

Answer 2

yes it is

Answer 3

so they want, like, \[2^1+2^0+2^{-1}+0+2^{-3}+\sum k^{something}2^{something}\]?
I don't think i get the question...

Answer 4

maybe take the difference of two infinite sums to get those terms...

Answer 5

@TuringTest yes that seems to be the gist. The given number can be written as
\[2^1+2^0+2^{-1}+2^{-3}+\sum_{k=4}^\infty2^{-k}\]
which can be summed nicely.

Answer 6

Oops, the series isn't exactly right, but you get the idea.

Answer 7

hm i have one smilar question with an answer it looks like \[0,\overline{1101}=\sum_{k=0}^{\infty} \frac{1}{2^{4k+1}}+\frac{1}{2^{4k+2}}+\frac{1}{2^{4k+4}}=...\]

Answer 8

yeah we gotta subtract of another series to get those 0's in there i think

Answer 9

\[\sum_{k=0}^\infty2^{2-k}-\sum_{k=0}^\infty2^{2-3k}\]?
something like that?

Answer 10

should be 4k\[\sum_{k=0}^\infty2^{2-k}-\sum_{k=0}^\infty2^{2-4k}\]not sure if that "explains" and I don't know how to write it as a "decimal fraction"

Answer 11

\[\large 11,101\overline{110} = 3.625+\sum_{k=0}^\infty2^{-(3k+4)} + 2^{-(3k+5)}\]

Answer 12

i think by decimal fraction they mean that sum for fractional part of binary number : 1/2 + 1/2^2 + ...

Answer 13

yes ganeshie exactly

Answer 14

note that you can evaluate that infinite sum as |r| = 1/8 < 1