Sum of series
Sum of Series
In mathematics, a series is the sum of the terms of a sequence of numbers. Given a sequence ${a_n}$, the sum of the first $n$ terms can be represented as $S_n = a_1 + a_2 + a_3 + \ldots + a_n$. The study of series is a major part of calculus and its applications can be found in various fields such as physics, engineering, and economics.
Types of Series
There are several types of series, but the most common ones are:
- Arithmetic Series
- Geometric Series
- Harmonic Series
- Power Series
Arithmetic Series
An arithmetic series is the sum of terms in an arithmetic sequence, where each term after the first is obtained by adding a constant difference.
Formula:
For an arithmetic series with first term $a_1$, common difference $d$, and $n$ terms, the sum is given by:
$$ S_n = \frac{n}{2}(2a_1 + (n-1)d) $$
Example:
Find the sum of the first 10 terms of the arithmetic series where $a_1 = 3$ and $d = 2$.
$$ S_{10} = \frac{10}{2}(2 \cdot 3 + (10-1) \cdot 2) = 5(6 + 18) = 5 \cdot 24 = 120 $$
Geometric Series
A geometric series is the sum of terms in a geometric sequence, where each term after the first is obtained by multiplying the previous term by a constant ratio.
Formula:
For a geometric series with first term $a_1$, common ratio $r$, and $n$ terms, the sum is given by:
$$ S_n = a_1 \frac{1 - r^n}{1 - r}, \quad r \neq 1 $$
Example:
Find the sum of the first 5 terms of the geometric series where $a_1 = 3$ and $r = 2$.
$$ S_5 = 3 \frac{1 - 2^5}{1 - 2} = 3 \frac{1 - 32}{-1} = 3 \cdot 31 = 93 $$
Harmonic Series
A harmonic series is the sum of the reciprocals of the terms of an arithmetic sequence.
Formula:
There is no simple closed-form formula for the sum of a harmonic series, but the sum of the first $n$ terms can be approximated using the natural logarithm:
$$ S_n \approx \ln(n) + \gamma $$
where $\gamma$ is the Euler-Mascheroni constant, approximately equal to $0.57721$.
Power Series
A power series is an infinite series of the form $\sum_{n=0}^{\infty} a_n x^n$, where $a_n$ represents the coefficient of the $n$-th term and $x$ is a variable.
Formula:
The sum of a power series can be represented as a function if it converges within its radius of convergence:
$$ f(x) = \sum_{n=0}^{\infty} a_n x^n $$
Table of Differences and Important Points
| Series Type | Formula for Sum of $n$ Terms | Example | Convergence Criteria |
|---|---|---|---|
| Arithmetic | $S_n = \frac{n}{2}(2a_1 + (n-1)d)$ | $S_{10} = 120$ for $a_1 = 3$, $d = 2$ | Always converges for finite $n$ |
| Geometric | $S_n = a_1 \frac{1 - r^n}{1 - r}$, $r \neq 1$ | $S_5 = 93$ for $a_1 = 3$, $r = 2$ | Converges if $ |
| Harmonic | No simple closed-form | $S_n \approx \ln(n) + \gamma$ | Diverges for infinite series |
| Power | $f(x) = \sum_{n=0}^{\infty} a_n x^n$ | Depends on coefficients $a_n$ and $x$ | Converges within radius of convergence |
Convergence and Divergence
The sum of an infinite series is said to converge if it approaches a finite limit as $n$ goes to infinity. Otherwise, it is said to diverge. The convergence of a series depends on the type of series and the values of its parameters.
For example, an infinite geometric series converges if the absolute value of the common ratio $|r|$ is less than 1. If $|r|$ is greater than or equal to 1, the series diverges.
Applications
Sum of series is a fundamental concept in mathematics with numerous applications:
- Calculating compound interest in finance.
- Analyzing harmonic vibrations in physics.
- Solving differential equations in engineering.
- Approximating functions using Taylor and Fourier series.
Understanding the sum of series is crucial for students and professionals in STEM fields, as it provides the tools to solve complex problems involving sequences and series.