Questions: Use sigma notation to write the sum. 1/(3(1)) + 1/(3(2)) + 1/(3(3)) + ... + 1/(3(7)) Σ from i=1 1/□

Solution Steps

To express the given sum in sigma notation, we identify the pattern in the terms. Each term is of the form \(\frac{1}{3i}\) where \(i\) ranges from 1 to 7. Therefore, the sum can be written as \(\sum_{i=1}^{7} \frac{1}{3i}\).

We are given the sum

\[ \frac{1}{3(1)} + \frac{1}{3(2)} + \frac{1}{3(3)} + \ldots + \frac{1}{3(7)} \]

This can be expressed in sigma notation as

\[ \sum_{i=1}^{7} \frac{1}{3i} \]

To calculate the sum, we evaluate

\[ \sum_{i=1}^{7} \frac{1}{3i} = \frac{1}{3} \sum_{i=1}^{7} \frac{1}{i} \]

The harmonic sum \(\sum_{i=1}^{7} \frac{1}{i}\) can be computed as

\[ 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} = \frac{363}{140} \]

Thus, we have

\[ \sum_{i=1}^{7} \frac{1}{3i} = \frac{1}{3} \cdot \frac{363}{140} = \frac{121}{140} \]

Calculating the decimal value gives us

\[ \frac{121}{140} \approx 0.8643 \]

Final Answer