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

Question

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

UpStudy AI · Accepted Answer

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.
Here are the series expressed in sigma notation: 1. $$ \sum_{k=1}^{6} 2k $$ 2. $$ \sum_{k=1}^{10} 3 \cdot (-2)^{k-1} $$ 3. $$ \sum_{k=1}^{7} \frac{1}{2^k} $$ 4. $$ \sum_{k=1}^{n} 5 $$ 5. $$ \sum_{k=1}^{5} k^3 $$ 6. $$ \sum_{k=1}^{n} k(k + 1) $$ 7. $$ \sum_{k=1}^{n} \frac{1}{k} $$ 8. $$ \sum_{k=1}^{24} \frac{k}{k + 1} $$

(d) Write each of the following series in sigma notation:

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