Question

The value of \[ \sum_{j=0}^{\infty} \sum_{i=1}^{\infty} 2^{-j} 3^{-i} \] is ________.

Hint:

When dealing with multiple summations or integrals with independent variables, you can often simplify the problem by separating them into a product of simpler expressions. For geometric series, pay close attention to the starting index (e.g., 0 or 1), as it determines the first term 'a' in the sum formula.

Answer 1

Correct Answer: 1

Step 1: Understanding the Question:
We need to evaluate a double summation which involves two infinite geometric series.
Step 2: Key Formula or Approach:
Since the indices 'j' and 'i' are independent, we can separate the double summation into a product of two single summations. \[ \sum_{j=0}^{\infty} \sum_{i=1}^{\infty} a_j b_i = \left(\sum_{j=0}^{\infty} a_j\right) \left(\sum_{i=1}^{\infty} b_i\right) \] We will use the formula for the sum of an infinite geometric series: $S = \frac{a}{1-r}$, where 'a' is the first term and 'r' is the common ratio, provided that $|r|<1$.
Step 3: Detailed Explanation:
Let's rewrite the expression and separate the sums: \[ \sum_{j=0}^{\infty} \sum_{i=1}^{\infty} (1/2)^{j} (1/3)^{i} = \left(\sum_{j=0}^{\infty} (1/2)^{j}\right) \times \left(\sum_{i=1}^{\infty} (1/3)^{i}\right) \] Now, we evaluate each sum separately. First Sum (for j): \[ S_j = \sum_{j=0}^{\infty} (1/2)^{j} = (1/2)^0 + (1/2)^1 + (1/2)^2 + \dots \] This is a geometric series with the first term $a = (1/2)^0 = 1$ and the common ratio $r = 1/2$. Using the sum formula: \[ S_j = \frac{a}{1-r} = \frac{1}{1 - 1/2} = \frac{1}{1/2} = 2 \] Second Sum (for i): \[ S_i = \sum_{i=1}^{\infty} (1/3)^{i} = (1/3)^1 + (1/3)^2 + (1/3)^3 + \dots \] This is a geometric series with the first term $a = (1/3)^1 = 1/3$ and the common ratio $r = 1/3$. Using the sum formula: \[ S_i = \frac{a}{1-r} = \frac{1/3}{1 - 1/3} = \frac{1/3}{2/3} = \frac{1}{2} \] Final Calculation: The value of the original double summation is the product of the two individual sums. \[ \text{Total Value} = S_j \times S_i = 2 \times \frac{1}{2} = 1 \] Step 4: Final Answer:
The value of the double summation is 1.