SERIESSUM in Google Sheets: Polynomial Coefficients and Series Steps

• Introduced: first service launch date is not established by the official history reviewed; no numbered desktop introduction is assigned.
• Availability: documented in the rolling Google Sheets service. Examples target ordinary web-spreadsheet cells; mobile clients and imported file modes are not separately certified.
• Checked: October 14, 2026; official SERIESSUM reference. Documentation and independently calculated examples, without native Sheets execution.

=SERIESSUM(x, n, m, a)

All four arguments are supplied: x is the input, n the first exponent, m the exponent step, and a the coefficient array or range. For coefficient position i starting at one, the term is a_i*x^(n+(i-1)*m).

Google’s argument prose describes m loosely as an increment to x, but its displayed series establishes that the increment belongs to the exponent. The input x stays fixed while exponents advance. This guide uses the displayed mathematical definition, with coefficients in one explicit vertical range.

Enter B2:B4 as 3, -2, 5. Enter 2 in A2 and put this formula elsewhere:

=SERIESSUM(A2,0,1,B2:B4)

The terms have powers 0, 1 and 2, giving contributions 3, -4, 20 and total 19. The separate arithmetic control =3-2*A2+5*A2^2 has the same expected result.

Changing A2 to 1 gives 6; changing it to -1 gives 10. These controls help reveal a sign mistake or a coefficient entered in the wrong row.

Enter 2, 0, 3 in D2:D4:

=SERIESSUM(2,0,1,D2:D4)

Expected result: 14, representing 2+0×2+3×2². Removing the zero and using only coefficients 2 and 3 with the same first exponent and step instead represents 2+3×2, giving 8.

An empty cell’s treatment is not established here. Use numeric zero for the missing coefficient so the polynomial’s intended positions are unambiguous.

With just coefficients 2, 3 in F2:F3, =SERIESSUM(2,0,2,F2:F3) also gives 14, because the step of two selects powers zero and two deliberately.

Enter coefficients 2, -1, 0.5 in H2:H4:

=SERIESSUM(2,1,2,H2:H4)

The exponents are 1, 3, 5. Contributions are 4, -8, 16, totaling 12. With x changed to -2, every selected power is odd and the expected total becomes -12.

Write the exponent sequence beside a custom model before entering coefficients. The same three numbers attached to consecutive powers would describe a different function, even though the syntax remained valid.

Enter J2:J6 as 1, 1, =1/2, =1/6, =1/24. These are reciprocal-factorial coefficients, deliberately entered as numeric formulas.

=SERIESSUM(0.5,0,1,J2:J6)
=EXP(0.5)

Independent expected values are 1.6484375 and approximately 1.648721271. The approximation is low by approximately 0.000283771.

Add =1/120 in J7 and extend the coefficient range to J2:J7. The series expectation becomes approximately 1.648697917, reducing the difference to approximately 0.000023354. This is a checked result at x=0.5, not a uniform accuracy promise for other inputs.

Google’s sample includes factorial values as coefficients. That is a valid custom series, but factorials are not the reciprocal-factorial coefficients of the exponential expansion used here. Choose coefficients for the intended model rather than copying a sample by appearance.

SERIESSUM evaluates only the coefficients supplied; it does not automatically generate more terms, estimate the remainder or prove convergence. A polynomial and a truncated infinite expansion may share syntax while requiring different interpretation.

Keep the coefficient range free of headings and unrelated totals. The selected reference does not provide a full native error/coercion matrix or every zero-base/negative-exponent edge case. Successful examples use finite numeric inputs with explicit coefficient positions.

For an ordinary total use SUM; for squared contributions use SUMSQ. The EXP comparison above is a separate function, not an instruction for SERIESSUM to calculate a remainder automatically.

• Google SERIESSUM help.
• Editorial testing standards.

Inputs are fictional. Approximate answers are independent mathematical expectations rounded to nine places, not captured native results. Complete type/error matrices, extreme-magnitude accuracy and cross-client behavior remain untested.