Associated Topics || Dr. Math Home || Search Dr. Math

### Infinite Product

```
Date: 12/01/97 at 02:13:53
From: Thang Nguyen
Subject: Infinite product

Hi, my name is Thang and I am having a hard time with this problem and
wondering if you can help me. My problem is how to find the infinite
product of this:

3^1/3 * 9^1/9 * 27^1/27 * ..... (3^n)^1/3^n

I have tried to solve for a general formula like the geometric series
where there is a general formula to work with, but that is where my

Thank you.
```

```
Date: 12/01/97 at 14:57:17
From: Doctor Bruce
Subject: Re: Infinite product

Hello Thang,

Try writing every term in your product as a power of 3.  So,

3^(1/3)  =  3^(1/3)
9^(1/9)  =  3^(2/9)
27^(1/27) =  3^(3/27),

and so forth. Then, your infinite product becomes

3^(1/3) * 3^(2/9) * 3^(3/27) * ...,

which can now be written as

3^[1/3 + 2/9 + 3/27 + ... ]

where the exponent is an infinite sum. Now the task becomes how to
evaluate this infinite sum. This can be done by writing it as an
infinite sum of infinite sums, like this

1/3 + 1/9 + 1/27 + 1/81 + ...   =  1/2
1/9 + 1/27 + 1/81 + ...   =  1/6
1/27 + 1/81 + ...   =  1/18
1/81 + ...   =  1/54
...   =  ...
__________________________________________

1/3 + 2/9 + 3/27 + 4/81 + ...   =  (1/2)[1 + (1/3) + (1/9) + ...  ]

=  3/4.

So it looks like the answer to your infinite product is  3^(3/4).

-Doctor Bruce,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```

```
Date: 12/11/97 at 16:02:51
From: Thang Nguyen
Subject: RE: calulus- infinite product

Hi Dr. Math,

I just want to say thank you so much for helping with this problem.

Thang Nguyen
```
Associated Topics:
High School Calculus
High School Sequences, Series

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search