Methods of Proof 357= 1 + 4
∑∞
i= 11
2 i∑∞
j= 01
2 j− 1 − 4
(
1 ·
1
2
+ 2 ·
1
4
+ 3 ·
1
8
+ 4 ·
1
16
+ 5 ·
1
32
)
= 9 −
65
8
=
7
8
< 1.
This is impossible, since the original value ofSwas 1. Consequently, there will always
be a pebble insideR, and this pebble will be at distance at most 5 from the origin.