(d) Write each of the following series in sigma notation: \( \begin{array}{ll}\text { (1) } 2+4+6+8+10+12 & \text { (2) } 3-6+12-24+\ldots \text { (to } 10 \text { terms) } \\ \text { (3) } \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots+\frac{1}{128} & \text { (4) } 5+5+5+5+\ldots \text { (to } n \text { terms) } \\ \text { (5) } 1+8+27+64+125 & \text { (6) } 2+6+12+20+\ldots \text { (to } n \text { terms) } \\ \text { (7) } 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\ldots \text { (to } n \text { terms) } & \text { (8) } \frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\frac{4}{5}+\ldots+\frac{24}{25}\end{array} \)

Certainly! Let's express each of the given series in sigma (summation) notation. ### (1) \(2 + 4 + 6 + 8 + 10 + 12\) This is an arithmetic series where each term increases by 2. The general term can be expressed as \(2k\), where \(k\) starts from 1. \[ \sum_{k=1}^{6} 2k \] --- ### (2) \(3 - 6 + 12 - 24 + \ldots\) (to 10 terms) This is a geometric series with alternating signs and each term being multiplied by \(-2\). The first term is 3. \[ \sum_{k=1}^{10} 3 \cdot (-2)^{k-1} \] --- ### (3) \(\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots + \frac{1}{128}\) This is a geometric series where each term is half of the previous term. The first term is \(\frac{1}{2}\) and the last term is \(\frac{1}{2^7}\). \[ \sum_{k=1}^{7} \frac{1}{2^k} \quad \text{or equivalently} \quad \sum_{k=1}^{7} 2^{-k} \] --- ### (4) \(5 + 5 + 5 + 5 + \ldots\) (to \(n\) terms) This is a constant series where the number 5 is added \(n\) times. \[ \sum_{k=1}^{n} 5 \] --- ### (5) \(1 + 8 + 27 + 64 + 125\) This series consists of the cubes of the first five natural numbers. \[ \sum_{k=1}^{5} k^3 \] --- ### (6) \(2 + 6 + 12 + 20 + \ldots\) (to \(n\) terms) Observing the pattern: - \(2 = 1 \times 2\) - \(6 = 2 \times 3\) - \(12 = 3 \times 4\) - \(20 = 4 \times 5\) The general term is \(k(k + 1)\). \[ \sum_{k=1}^{n} k(k + 1) \] --- ### (7) \(1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots\) (to \(n\) terms) This is the harmonic series up to \(n\) terms. \[ \sum_{k=1}^{n} \frac{1}{k} \] --- ### (8) \(\frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} + \ldots + \frac{24}{25}\) Each term has the form \(\frac{k}{k + 1}\), starting from \(k = 1\) up to \(k = 24\). \[ \sum_{k=1}^{24} \frac{k}{k + 1} \] --- These sigma notations concisely represent each of the given series, capturing their patterns and limits.