SOLUTION: what is the value of 9^(1/3) x 9^(1/9) x 9^(1/27) x ...? also is there a general formula in solving these types of question? thanks!

Question 1107236: what is the value of 9^(1/3) x 9^(1/9) x 9^(1/27) x ...?
also is there a general formula in solving these types of question? thanks!
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
You know that .
So,
,
because

That sum is the sum of an infinite geometric sequence.
The "formula" for such a sum with a first term and a common ratio is
.
In the case of the sum above, with and ,
.

There is no need to memorize formulas, if you can deduce them.
Consider the sum or terms
.
, and
--> --> .
If , we might like to write it as
,
and as approaches , approaches ,
and approaches .
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