How do you find the sum of the infinite geometric series 8 + 4 + 2 + 1 +...?

1 Answer
Mar 3, 2018

#16#

Explanation:

This converging geometric progression is a special one.
#"G.P " 8,4,2,1,1/2,1/4...#
Common multiple is #1/2# .

When you find the sum of this G.P
#=>8+4+2+1+1/2+1/4....#

#=>15+1/2+1/4+1/8.....#

#=>15+1#

Basically, you need to prove #1/2+1/4+1/8...=1#

Proof:
#S_n=1/2+1/4+...1/2^(n-1)#

Multiply #2# both sides

#2S_n=2/2+2/4+...1/2^n#

Subtract #S_n# from this.

#2S_n-S_n=1+2/4-1/2+2/8-1/4....#

#S_n=1-1/2^n#

As you increase the value of #n# the value of #S_n# tends to be #1#.

There's one more proof which is easy to understand and really amazing.

Given below is a square of side 1unit. First cut it half, then #(1/4 )^(th)# and then #(1/8) ^(th)# and so on. You'll see that as you increase the value of #n# the sum tends to be #1#.

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