2026-09-29 9 min

Power Series Calculator

Evaluate a polynomial-style power series sum a(k)·(x−c)^k at x with Horner-like accumulation plus a ratio radius note.

Calculator

Power Series Inputs

Center (c)

Point (x)

Coefficients a(0), a(1), ... (comma separated)

a(0) first, up to 12 numbers. Parsed: 4

Results update live as you type

Status:P(1) = 2.6667OUTSIDE hint R≈1

Power Series Summary

P(x)2.6667

TrustOutside hint

Setupc=0, x=1

Term list

1, 1, 0.5, 0.1667

Bars show each term\'s contribution; the running line accumulates them.

Results

Caution

P(x) Value

2.6667

Truncated sum

Displacement t = x−c

Power base

Radius Hint R

min |a(k)/a(k+1)|

Inside Hint?

No — suspect

|t| vs R

Last Term

0.1667

Truncation tail

Terms Used

Parsed coefficients

Horner-like accumulation on t = x−c; radius hint from adjacent coefficient ratios with inside/outside flag.

Visualizations

Series term contributions

1a1

1a2

0.5a3

0.2a4

Accumulation toward P(x)

Terms used

count / 124

P(x) vs |displacement|

P(x)2.7

Formula

Powers of the displacement fan out from the center, weighted by your coefficients — Horner-style accumulation keeps it stable.

P(x) = a(0) + a(1)(x−c) + ... + a(m-1)(x−c)^(m-1)

Alternative Form

Radius hint: R ≈ min |a(k)/a(k+1)| over usable pairs

Variables

Symbol	Name	Unit	Description
a(k)	Coefficients	number	Comma list, a(0) first
c	Center	number	Expansion point
x	Evaluation point	number	Where to sum
R	Radius hint	number	Adjacent-ratio estimate

Explanation

Near the center, high powers vanish and few terms suffice; far away, they explode — the radius marks the border.

Derivation

Direct Horner-like loop on t = x − c. Ratio hint from |a(k)/a(k+1)| per the ratio test on series terms.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Parse coefficients in a(0)-first order.

Split on commas

coeffs:1, 1, 0.5, 0.1667

c:0

x:1

Result: 4 coefficients parsed

Step 2

Form the displacement from the center.

t = x − c

Intermediate: 1 − 0 = 1

Result: t = 1

Step 3

Accumulate term by term.

term(k) = a(k)·t^k

Intermediate: 1, 1, 0.5, 0.1667

Result: P = 2.6667

Step 4

Note the truncation tail.

last term magnitude

Intermediate: 0.1667

Result: tail ≈ 0.1667

Step 5

Estimate the radius from neighbors.

R ≈ min|a(k)/a(k+1)|

Intermediate: R ≈ 1

Result: R ≈ 1

Step 6

Judge trust by displacement vs radius.

|t| < R ?

Intermediate: |1| vs 1

Result: Outside — treat as suspect

Final Answer

2.6667

Worked Examples

See how the calculator works with different scenarios.

Scenario

c=0, x=1, 4 coeffs.

Inputs

coeffs:1, 1, 0.5, 0.1667

c:0

x:1

Calculation

P = 1+1+0.5+0.1667 = 2.6667

Result

P(1) ≈ 2.6667

Interpretation

Four terms already near e ≈ 2.718.

Understanding Your Result

Your Result: P(x) for your coefficients, center, and x — see the value card.

Typical range: Up to 12 coefficients parsed; |x − c| vs R hint judges trust.

What This Means

The truncated power-series value at x around c.

Why It Matters

Taylor polynomials, generating functions, and signal filters all evaluate this exact sum.

Professional Interpretation

If |x − c| exceeds the hint, the page warns the truncation may mislead — radius discipline built in.

Recommendations

•List a(0) first
•Compare |x−c| with R hint
•Use more terms near the edge
•Verify constant series returns a(0)
•Watch alternating cancellation

Educational Content

Introduction

Feed the weights, pick where to stand, and add up the powers — centered arithmetic at its cleanest.

Concept Explanation

Centered monomials as a basis: displacement powers scaled by coefficients.

Applications

▸Taylor polynomial evaluation

▸Generating-function queries

▸Filter tap summation

▸Curve fitting checks

▸Physics perturbation sums

▸Probability moments

Benefits

•Comma-list input
•Horner-like stability
•Radius hint included
•Term-by-term trace
•Center-agnostic math

Limitations

•Truncated sums only
•Hint needs 2+ nonzero neighbors
•12-coefficient parse cap

Best Practices

•Order a(0) first
•Count terms used
•Check displacement size
•Compare hint vs distance
•Verify tiny displacements

Common Misconceptions

Myth: More terms always help.

Truth: Outside the radius, partials can worsen — check |x−c| < R.

Myth: Center does not matter.

Truth: It anchors every power: (x−c)^k shifts the whole basis.

Myth: Radius hint is exact.

Truth: It is a neighbor-ratio heuristic for truncated lists.

Myth: Coefficient order is free.

Truth: a(0) must lead; reversal evaluates a different series.

Practical Tips

•Enter a(0) first, comma separated
•Compare displacement with the R hint
•Add terms when near the edge
•Verify constants return a(0)
•Keep coefficients finite
•Quote value with center
•Re-check order after edits

Common Mistakes

Reversing coefficient order

Lead with a(0); last entry is the highest power.

Evaluating far outside R

Recenter or accept divergence — do not trust the number.

Mixing x with (x−c)

Powers use displacement t = x − c, not raw x.

Reading R hint as guarantee

Treat as guidance from few ratios.

Pasting non-numbers

Non-numeric tokens are ignored; check the parsed count.

Assumptions

This calculator makes the following assumptions to provide estimates:

Listed order: a(0) first through highest last.

Truncation: Sum covers parsed terms only.

Real arithmetic: c, x real.

Double powers: t^k in doubles.

Local only: Browser-side.

Limitations

Be aware of these limitations when using this calculator:

Finite truncation: True infinite behavior needs theory.

Heuristic radius: Few ratios, not a proof.

Parse cap: 12 coefficients max.

Cancellation noise: Alternating far sums round.

Frequently Asked Questions

Answers to common questions about this calculator.

How do I enter coefficients?

Comma separated, a(0) first: e^x ≈ 1,1,0.5,0.1667,...

Example e^1?

c=0, x=1, coeffs 1,1,0.5,0.1667,0.0417 → ≈ 2.708 (5 terms).

What is the center?

Where displacements start: t = x − c. Maclaurin means c = 0.

What is the radius hint?

Smallest usable |a(k)/a(k+1)| — believable inside, suspect outside.

Why Horner-like?

Nesting t-factors reduces rounding vs raw powers.

How many terms?

Up to 12 parsed; near-center needs few, near-edge needs many.

Alternating series?

Signs ride in coefficients; cancellation can cost digits far out.

Constant series?

Single a(0) returns a(0) everywhere.

Divergence?

Beyond R the infinite series fails — truncation is then decorative.

Stored?

No.

Related Calculators

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Related Guides

Deepen your understanding with these educational resources.

Sequences vs series in one sketch

Terms, sums, and convergence at a glance.

Closed forms that skip the loop

Jump straight to term n without listing.

Growth regimes: linear vs exponential

Reading difference and ratio correctly.

Checking sums by pairing and ratio tricks

Endpoint pairing and the r-times-Sn proof.

Glossary

Technical terms used in mortgage calculations explained in plain language.

Power series: Sum a(k)(x−c)^k.
Coefficient: Weight a(k) of the kth power.
Center: c anchoring displacements.
Displacement: t = x − c.
Truncation: Finite prefix of the series.
Radius of convergence: R bounding reliable x.
Ratio hint: min |a(k)/a(k+1)| estimate.
Horner scheme: Nested evaluation order.
Monomial basis: (x−c)^k building blocks.

Authoritative References

Trusted sources and standards referenced by this calculator.

Wikipedia — Sequence Wikipedia — Series (mathematics) Khan Academy — Sequences Math is Fun — Sequences and Series Purplemath — Sequences and Series Wolfram MathWorld — Sequence Cuemath — Sequences and Series

Applicable Standards

Common Core HSF-BF.A — building sequences and seriesNCTM Number and Operations — patterns, functions, and series

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Page Information

Last Updated:

2026-09-29

Version:

2.0.0

References:

License:

Free to use for educational and professional purposes.