Write each of the following scries in sigma notation: $$ \begin{array}{ll} ext { (a) } 2+4+6+8+10+12 & ext { (b) } 1+8+27+64+125 \ ext { (c) } \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots .+\frac{1}{128} & ext { (d) } 3-6+12-24+\ldots . ext { to } n ext { terms } \ ext { (c) } \frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\frac{4}{5}+\ldots .+\frac{24}{25} & ext { (f) } 5+5+5+5+5+\ldots . ext { to } n ext { terms } \ ext { (g) } 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\ldots . & ext { (The Harmonic Series) }\end{array} $$

Question

Write each of the following scries in sigma notation: $$ \begin{array}{ll}	ext { (a) } 2+4+6+8+10+12 & 	ext { (b) } 1+8+27+64+125 \ 	ext { (c) } \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots .+\frac{1}{128} & 	ext { (d) } 3-6+12-24+\ldots . 	ext { to } n 	ext { terms } \ 	ext { (c) } \frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\frac{4}{5}+\ldots .+\frac{24}{25} & 	ext { (f) } 5+5+5+5+5+\ldots . 	ext { to } n 	ext { terms } \ 	ext { (g) } 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\ldots . & 	ext { (The Harmonic Series) }\end{array} $$

UpStudy AI · Accepted Answer

Certainly! Below are the sigma notation representations for each of the given series:

### (a) $$ 2 + 4 + 6 + 8 + 10 + 12 $$

This is an arithmetic series where each term increases by 2. It can be expressed as:

$$
\sum_{i=1}^{6} 2i
$$

### (b) $$ 1 + 8 + 27 + 64 + 125 $$

These terms are cubes of integers. The series can be written as:

$$
\sum_{i=1}^{5} i^3
$$

### (c) $$ \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots + \frac{1}{128} $$

This is a geometric series with the first term $$ \frac{1}{2} $$ and a common ratio of $$ \frac{1}{2} $$. It has 7 terms (since $$ \frac{1}{128} = \frac{1}{2^7} $$):

$$
\sum_{i=1}^{7} \frac{1}{2^i}
$$

### (d) $$ 3 - 6 + 12 - 24 + \ldots $$ (to $$ n $$ terms)

This is a geometric series with the first term 3 and a common ratio of -2. The sigma notation is:

$$
\sum_{i=1}^{n} 3 \cdot (-2)^{i-1}
$$

### (e) $$ \frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} + \ldots + \frac{24}{25} $$

Each term can be represented as $$ \frac{i}{i+1} $$ for $$ i $$ from 1 to 24:

$$
\sum_{i=1}^{24} \frac{i}{i+1}
$$

### (f) $$ 5 + 5 + 5 + 5 + 5 + \ldots $$ (to $$ n $$ terms)

This is a constant series where each term is 5. The sigma notation is:

$$
\sum_{i=1}^{n} 5
$$

### (g) $$ 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots $$ (The Harmonic Series)

The harmonic series is an infinite series where each term is the reciprocal of an integer:

$$
\sum_{i=1}^{\infty} \frac{1}{i}
$$
Here are the sigma notations for each series: (a) $$ \sum_{i=1}^{6} 2i $$ (b) $$ \sum_{i=1}^{5} i^3 $$ (c) $$ \sum_{i=1}^{7} \frac{1}{2^i} $$ (d) $$ \sum_{i=1}^{n} 3 \cdot (-2)^{i-1} $$ (e) $$ \sum_{i=1}^{24} \frac{i}{i+1} $$ (f) $$ \sum_{i=1}^{n} 5 $$ (g) $$ \sum_{i=1}^{\infty} \frac{1}{i} $$

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