Вивчайте Introduction to Series

A series is the sum of the terms of a sequence. Two common types of series are arithmetic and geometric series.

Arithmetic Series

An arithmetic series is formed when the difference between consecutive terms in a sequence is constant.

2, 5, 8, 11, 14, ...; (\text{common difference}, d = 3)

The sum of the first $n$ terms of an arithmetic series is given by:

S_n = \frac{n}{2} \cdot (a + l)

Where:

$n$ - number of terms;
$a$ - first term;
$l$ - last term.

Alternatively, if the last term $l$ is not known:

S_n = \frac{n}{2} \cdot 2a + (n - 1) \cdot d

Example

Find the sum of the first 10 terms of the series $2,5,8,...$

S_{10} = \frac{10}{2} \cdot (2 + (10 - 1) \cdot 3) = 5 \cdot (2 + 27) = 145

Geometric Series

A geometric series is formed when each term in the sequence is multiplied by a fixed ratio to get the next term.

3,6,12,24,48,...;(\text{common ratio}, r=2)

The sum of the first $n$ terms of a geometric series is given by:

S_n = a \cdot \frac{1 - r^n}{1 - r},\ r \neq 1

Where:

$a$ - first term;
$r$ - common ratio;
$n$ - number of terms.

If the series is infinite and $|r|<1$ :

S = \frac{a}{1 - r}

Example:

Find the sum of the first 4 terms of the series $3,6,12,24,...$

S_4 = 3 \cdot \frac{1-2^4}{1-2} = 3 \cdot \frac{1-16}{-1}=3 \cdot 15 = 45

Real-World Applications

Arithmetic and geometric series have a wide range of applications in data science and beyond:

Population growth: geometric series can model exponential growth in populations or resource use;
Financial modeling: geometric series are used to calculate compound interest and investment returns;
Revenue prediction: summing revenue across time periods or forecasting future earnings;
Machine learning: some algorithms, such as gradient descent, involve summations that resemble arithmetic or geometric series.

1. What is the sum of the first 5 terms of the series $3,6,9,12,...$ ?

2. $a=1$ , $r=0.5$ and $n=\infty$ , what is the sum of the infinite geometric series?

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A series is the sum of the terms of a sequence. Two common types of series are arithmetic and geometric series.

Arithmetic Series

An arithmetic series is formed when the difference between consecutive terms in a sequence is constant.

2, 5, 8, 11, 14, ...; (\text{common difference}, d = 3)

The sum of the first $n$ terms of an arithmetic series is given by:

S_n = \frac{n}{2} \cdot (a + l)

Where:

$n$ - number of terms;
$a$ - first term;
$l$ - last term.

Alternatively, if the last term $l$ is not known:

S_n = \frac{n}{2} \cdot 2a + (n - 1) \cdot d

Example

Find the sum of the first 10 terms of the series $2,5,8,...$

S_{10} = \frac{10}{2} \cdot (2 + (10 - 1) \cdot 3) = 5 \cdot (2 + 27) = 145

Geometric Series

A geometric series is formed when each term in the sequence is multiplied by a fixed ratio to get the next term.

3,6,12,24,48,...;(\text{common ratio}, r=2)

The sum of the first $n$ terms of a geometric series is given by:

S_n = a \cdot \frac{1 - r^n}{1 - r},\ r \neq 1

Where:

$a$ - first term;
$r$ - common ratio;
$n$ - number of terms.

If the series is infinite and $|r|<1$ :

S = \frac{a}{1 - r}

Example:

Find the sum of the first 4 terms of the series $3,6,12,24,...$

S_4 = 3 \cdot \frac{1-2^4}{1-2} = 3 \cdot \frac{1-16}{-1}=3 \cdot 15 = 45

Real-World Applications

Arithmetic and geometric series have a wide range of applications in data science and beyond:

Population growth: geometric series can model exponential growth in populations or resource use;
Financial modeling: geometric series are used to calculate compound interest and investment returns;
Revenue prediction: summing revenue across time periods or forecasting future earnings;
Machine learning: some algorithms, such as gradient descent, involve summations that resemble arithmetic or geometric series.

1. What is the sum of the first 5 terms of the series $3,6,9,12,...$ ?

2. $a=1$ , $r=0.5$ and $n=\infty$ , what is the sum of the infinite geometric series?

Все було зрозуміло?

Дякуємо за ваш відгук!

Секція 2. Розділ 4

Introduction to Series

Arithmetic Series

Example

Geometric Series

Example:

Real-World Applications

1. What is the sum of the first 5 terms of the series 3,6,9,12,...3,6,9,12,...3,6,9,12,...?

2. a=1a=1a=1, r=0.5r=0.5r=0.5 and n=∞n=\inftyn=∞, what is the sum of the infinite geometric series?

Awesome!

Introduction to Series

Arithmetic Series

Example

Geometric Series

Example:

Real-World Applications

1. What is the sum of the first 5 terms of the series 3,6,9,12,...3,6,9,12,...3,6,9,12,...?

2. a=1a=1a=1, r=0.5r=0.5r=0.5 and n=∞n=\inftyn=∞, what is the sum of the infinite geometric series?

1. What is the sum of the first 5 terms of the series $3,6,9,12,...$ ?

2. $a=1$ , $r=0.5$ and $n=\infty$ , what is the sum of the infinite geometric series?

1. What is the sum of the first 5 terms of the series $3,6,9,12,...$ ?

2. $a=1$ , $r=0.5$ and $n=\infty$ , what is the sum of the infinite geometric series?