Help in multiple sigma question

Question

Find the value of $\displaystyle \underset{(i \ne j \ne k)}{\sum_{i=0}^\infty \sum_{j=0}^\infty \sum_{k=0}^\infty \frac 1{3^i 3^j 3^k}}$.

How to solve questions containing multiple sigma like this question? How can i solve such questions please help. This is question is just in general actually i want help to encounter questions with multiple sigma.

It's answer is $81/208$

Answer 1

Hint: use the inclusion-exclusion principle: $$\displaystyle \underset{(i \ne j \ne k)}{\sum_{i=0}^\infty \sum_{j=0}^\infty \sum_{k=0}^\infty \frac 1{3^i 3^j 3^k}}=\cdots=\left(\frac32\right)^3-3\cdot\left(\frac32\cdot \frac98\right)+2\cdot\frac{27}{26}=\frac{81}{208}.$$

Answer 2

I guess you meant $i \neq j, j \neq k, k \neq i$. One of such case is $i < j < k$. In that case, $$ \sum^\infty_{i=0} \sum^\infty_{j=i+1} \sum^\infty_{k=j+1} \frac{1}{3^i 3^j 3^k}$$ is a part of given expression. Now, how much there are such cases?