Question

Show that the series S=1-1+1-1+1-1... can be assigned any real value x by re-bracketing and exploiting the fact that sometimes S=0 and sometimes S=1.

Answer 1

If the number of items is even then
S = +1-1 +1-1....
S = [+1-1] +[1-1] ....[1-1]
S = [0]+[0]+[0] ....
Then S = 0

if the number of items is odd
S = [+1-1] +[1-1] ....[1-1]+[1 = 1

Then S=1

Answer 2

How can this be used to show that S can be any real value, such as decimals, transcendentals (e,pi)?

Answer 3

well S=1 so S is a real value do to it can be a decimal, transcendentals