The meaning of the decimal representation of a number 0.d1d2d3 . . . (where the digit i is one of the numbers 0, 1, 2, . . ., 9) is that 0.d1d2d3d4 . . . = d1/10 + d2/10^2 + d3/10^3 + d4/10^4 + . . . Show that this series always converges.

Question

The meaning of the decimal representation of a number 0.d1d2d3 . . . (where the digit i is one of the numbers 0, 1, 2, . . ., 9) is that 0.d1d2d3d4 . . . = d1/10 + d2/10^2 + d3/10^3 + d4/10^4 + . . .    Show that this series always converges.

anonymous · Answer

Geeze, what grade is this?? ((I do not know the answer.))

anonymous · Answer

This is calculus 2, so its gonna be tricky!

myininaya · Answer

so you are having trouble this converges: sum(1/10^i,i=1..n)?
if this sum coverges then sum(di/10^i,i=1..n) should converges i think

myininaya · Answer

ratio test
we can show it converges absolutely 
lim          |d{n+1}/10^(n+1)|/|d{n}/10^n|
n->inf
=  "          |d{n+1}/10^n*10|*|10^n/d{n}|
=  "          1/10*|d{n+1}/d{n}|
the digits can range form 0 to 9
the biggest number that can be in that absolute value thing is 9/1
so
1/10*9=9/10<1
so it converges absolutely

anonymous · Answer

thank you!  this helps immensely.  I had a solution worked out involving the "r" value being less than one for 9/10^n (geometric series), making it convergent

anonymous · Answer

r being 1/10