(d) Write each of the following series in sigma notation (1) \( 2+4+6+8+10+12 \) (3) \( \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\text{...}+\frac{1}{128} \) (5) \( 1+8+27+64+125 \) (7) \( 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\text{...} \) (to \( n \) terms)

Certainly! Let's express each of the given series in **sigma notation**: --- ### (1) \( 2 + 4 + 6 + 8 + 10 + 12 \) **Analysis:** - **Pattern:** Each term increases by 2, which can be represented as \(2k\), where \(k\) starts from 1. - **Number of Terms:** There are 6 terms. **Sigma Notation:** \[ \sum_{k=1}^{6} 2k \] --- ### (3) \( \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots + \frac{1}{128} \) **Analysis:** - **Pattern:** Each term is a reciprocal of a power of 2. Specifically, the terms are \( \frac{1}{2^k} \). - **Number of Terms:** The last term is \( \frac{1}{128} = \frac{1}{2^7} \), so there are 7 terms. **Sigma Notation:** \[ \sum_{k=1}^{7} \frac{1}{2^k} \] *Alternatively, you can write:* \[ \sum_{k=1}^{7} \left( \frac{1}{2} \right)^k \] --- ### (5) \( 1 + 8 + 27 + 64 + 125 \) **Analysis:** - **Pattern:** Each term is a perfect cube. Specifically, the terms are \( k^3 \) where \(k\) starts from 1. - **Number of Terms:** There are 5 terms. **Sigma Notation:** \[ \sum_{k=1}^{5} k^3 \] --- ### (7) \( 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots \) (to \( n \) terms) **Analysis:** - **Pattern:** This is known as the **harmonic series**, where each term is \( \frac{1}{k} \). - **Number of Terms:** The series continues up to the \( n \)-th term. **Sigma Notation:** \[ \sum_{k=1}^{n} \frac{1}{k} \] --- These sigma notations succinctly capture the essence and pattern of each given series.