Find the sum of the infinite series 0.9 + 0.09 + 0.009 +...if it exists.

Question

aum · Answer

This is a geometric series.  Can you figure out the common ratio?

anonymous · Answer

0.1

aum · Answer

Yes.  First term a = 0.9.  Plug it into the formula for an infinite geometric series.

aum · Answer

Plug it into the formula for the sum of an infinite geometric series.

ikram002p · Answer

0.9999999999999999999999999999999999999999999999999999....= ?

anonymous · Answer

1

aum · Answer

Yes.

ikram002p · Answer

:) the same as ur series

kainui · Answer

What is this, just a repeating decimal, .9 repeating.$$\LARGE x=.9999...$$ multiply both sides of this equation by 10$$\LARGE x10=9.9999...$$ subtract the first one from the second one $$\LARGE x10-x=(9.9999...)-(.99999...) $$ Now we are just left with this$$\LARGE 9x=9$$ Oh hey, wow.$$\LARGE x=1$$

anonymous · Answer

thank yall!

aum · Answer

You are welcome.  Three different approaches to the same problem yielding the same solution! :)

kainui · Answer

In fact, the difference is purely superficial. They are all really the same approach disguised slightly. The formula for the geometric series is derived in this same sort of way. Although it is nice to see the famous 1=.9999... formula shown to be true!