SERIESSUM in Excel: Build Power Series with Explicit Coefficients

• Introduced: First introduction version is not established by the official history reviewed.
• Availability: official help lists Excel for Microsoft 365, Excel for Microsoft 365 for Mac, Excel 2024, Excel 2024 for Mac, Excel 2021, Excel 2021 for Mac, Excel 2019, Excel 2016.
• Checked: October 14, 2026; official SERIESSUM reference. Documentation and independently calculated examples; no native Excel execution or broader platform certification.

=SERIESSUM(x, n, m, coefficients)

All four arguments are required. The first exponent is n, successive exponents increase by m, and the number of coefficient values determines how many terms are added. For coefficients a₁, a₂ and a₃, the expression is a₁xⁿ+a₂x⁽ⁿ⁺ᵐ⁾+a₃x⁽ⁿ⁺²ᵐ⁾.

Microsoft specifies #VALUE! for a nonnumeric argument. The successful controls use numeric coefficients and finite real powers. Fractional exponents with negative bases, zero raised to problematic powers and complete imported-type behavior are not certified here.

Put coefficients 3, -2, 5 in A2:A4 and numeric 2 in C2:

=SERIESSUM(C2,0,1,A2:A4)
=3-2*C2+5*C2^2

Both have expected result 19. The coefficient order matters: the first multiplies the constant term, the second the first power, and the third the square. Reversing the coefficient column changes the polynomial rather than merely sorting its inputs.

At x=1, this polynomial has expected result 6. At x=-1, it has expected result 10, because the linear term changes sign while the squared term does not. Use these contrasting inputs to detect a misplaced coefficient.

To represent 2+3x² using exponent step one, enter coefficients 2, 0, 3 in E2:E4:

=SERIESSUM(2,0,1,E2:E4)

Expected result: 14, from 2+0×2+3×4. Omitting the zero and supplying only 2 and 3 with the same exponent settings would represent 2+3x instead and give 8 at x=2.

Alternatively, coefficients 2 and 3 with starting exponent zero and step two also represent 2+3x². The choice of coefficient positions and exponent step must be considered together.

Enter coefficients 2, -1, 0.5 in G2:G4:

=SERIESSUM(2,1,2,G2:G4)

The powers are 1, 3 and 5, so the expected result is 12, from 2×2-1×8+0.5×32. The step is added to the exponent; it does not multiply the x value or skip coefficient cells.

At x=-2, the corresponding expected result is -12, since every exponent in this particular series is odd. That sign symmetry would not hold for an arbitrary mixture of odd and even powers.

Use x=0.5 and coefficients 1, 1, 1/2, 1/6, 1/24 for exponents zero through four. In I2:I6 enter formulas =1, =1, =1/2, =1/6, =1/24, then calculate:

=SERIESSUM(0.5,0,1,I2:I6)
=EXP(0.5)

Expected results are approximately 1.648437500 and 1.648721271. Their difference is approximately 0.000283771. This is a finite truncation of a known exponential series, not an assertion that SERIESSUM computes EXP exactly.

Adding the next coefficient =1/120 in I7 and extending the range to I2:I7 gives approximately 1.648697917, reducing the difference to approximately 0.000023354. The additional term has a predictable mathematical role; more terms do not guarantee improved accuracy for every arbitrary series or input.

Check the coefficient range length, its order and the first exponent before comparing results with another function. Typing a rounded decimal approximation to 1/6 instead of the formula =1/6 introduces a separate coefficient-rounding error.

The finite polynomial model is exact symbolically, while a truncated infinite series differs from its target function even if every arithmetic step is correct. Document the input range and acceptable error for an approximation before reusing it beyond the demonstration value.

A missing zero coefficient shifts later exponents when the step is one. A wrong starting exponent can alter every term. Inspect the first few expanded terms in helper cells when a compact formula looks plausible but disagrees with the expected value.

No native Excel calculation was run. Large exponents, convergence outside the chosen input and exhaustive coercion/error behavior require separate evidence; the small polynomial and approximation controls do not certify them.

• Microsoft SERIESSUM reference.
• Editorial testing standards.

Worked inputs are fictional. Decimal answers are independently calculated mathematical expectations rounded to nine places, not captured native results. Extreme magnitudes, complete coercion matrices and cross-version numerical accuracy remain untested.