SERIESSUM function

The SERIESSUM function in Excel calculates the sum of a power series based on the formula:

$$\text{SERIESSUM} = a_1 \cdot x^{n} + a_2 \cdot x^{n+m} + a_3 \cdot x^{n+2m} + \dots$$

This is useful in mathematical modeling and calculations involving power series expansions.

Syntax

=SERIESSUM(x, n, m, coefficients)

Parameters

1. x (required): The input value for the series (the variable in the equation).
2. n (required): The initial power to which the value of x is raised.
3. m (required): The step by which the power increases in each term.
4. coefficients (required): An array or range of coefficients for each term in the series.

How It Works

1. The first term is $\text{coefficients}[1] \cdot x^n$.
2. The second term is $\text{coefficients}[2] \cdot x^{n+m}$.
3. The third term is $\text{coefficients}[3] \cdot x^{n+2m}$.
4. The series continues for all coefficients provided.

Examples

Example 1: Simple Series

To calculate the series:

$$2 \cdot x^0 + 3 \cdot x^1 + 4 \cdot x^2$$

If $x = 2$, $n = 0$, $m = 1$, and coefficients are in cells A1:A3 (2, 3, 4):

=SERIESSUM(2, 0, 1, A1:A3)

Result: 26

Example 2: Exponential Approximation

To approximate $e^x$ using the series:

$$1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!}$$

If $x = 1$, $n = 0$, $m = 1$, and coefficients $1, 1/1!, 1/2!, 1/3!$ are in cells A1:A4:

=SERIESSUM(1, 0, 1, A1:A4)

Result: 2.6667 (approximation for $e^1$).

Notes

• Coefficients: Must be provided as an array or range. The length of the array determines how many terms are included in the calculation.
• Numeric Input: The x, n, m, and coefficients must be numeric. Non-numeric inputs will return #VALUE!.
• Practical Use: Often used in Taylor series, Fourier series, or other expansions in mathematics.

Related Functions

• SUM: Adds numbers together but does not involve powers.
• POWER: Calculates a number raised to a specific power.
• EXP: Calculates $e^x$, the base of natural logarithms raised to a power.

The SERIESSUM function is powerful for calculations involving polynomial expansions, approximations, and mathematical series.