Series Calculator
Calculate arithmetic series, geometric series, power series, and test for convergence with detailed step-by-step solutions.
Arithmetic Series
An arithmetic series is the sum of an arithmetic sequence where each term differs by a constant value.
Result
Step-by-Step Solution
Series Terms
Quick Examples:
Understanding Series
A series is the sum of the terms of a sequence. Series are fundamental in calculus and analysis, with applications in physics, engineering, and many areas of mathematics.
Types of Series
- Arithmetic Series: Sum of arithmetic sequence where consecutive terms have a constant difference.
- Geometric Series: Sum of geometric sequence where consecutive terms have a constant ratio.
- Power Series: Series of the form ∑aₙxⁿ, important in calculus and analysis.
- Taylor Series: Power series representation of functions around a point.
- Fourier Series: Representation of periodic functions as sums of sines and cosines.
Series Formulas
Arithmetic Series
\[ S_n = \frac{n}{2}[2a + (n-1)d] = \frac{n}{2}(a_1 + a_n) \]
where a = first term, d = common difference, n = number of terms
Finite Geometric Series
\[ S_n = a \cdot \frac{1-r^n}{1-r} \quad (r \neq 1) \]
where a = first term, r = common ratio, n = number of terms
Infinite Geometric Series
\[ S = \frac{a}{1-r} \quad (|r| < 1) \]
Converges only when |r| < 1
Convergence Tests
| Test | Condition | Result |
|---|---|---|
| Ratio Test | $$ \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = L $$ | L < 1: converges, L > 1: diverges |
| Root Test | $$ \lim_{n \to \infty} \sqrt[n]{|a_n|} = L $$ | L < 1: converges, L > 1: diverges |
| Comparison Test | Compare with known series | If 0 ≤ aₙ ≤ bₙ and ∑bₙ converges, then ∑aₙ converges |
| Alternating Series | aₙ > 0, decreasing, lim aₙ = 0 | ∑(-1)ⁿaₙ converges |
| p-Series Test | $$ \sum \frac{1}{n^p} $$ | Converges if p > 1, diverges if p ≤ 1 |
Famous Series
Basel Problem:
$$ \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} $$
Exponential Series:
$$ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots $$
Geometric Series:
$$ \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + \cdots \quad (|x| < 1) $$
Applications of Series
Series are used in numerical analysis for approximating functions, in physics for modeling wave phenomena, in engineering for signal processing, and in computer science for algorithms and data compression.
Series Calculator Answers
What does the series calculator do?
The series calculator computes the sum of a sequence of numbers, whether it's arithmetic, geometric, or based on a custom formula. It helps evaluate partial sums or full series values.
Which types of series can it handle?
It supports arithmetic series, geometric series, infinite series (if convergent), and series based on user-defined general terms. Some tools also handle power or Taylor series.
Can I use it to find the sum of a certain number of terms?
Yes. Just enter the number of terms (n) and the rule or values of the sequence, and it will return the partial sum up to the nth term.
How is this helpful for students?
It’s great for checking homework, learning how series behave, and understanding summation notation in algebra and calculus courses. It also aids in exploring convergence.