2026-09-29 9 min

Maclaurin Series Calculator

Approximate e^x, sin x, cos x, ln(1+x) by Maclaurin polynomials (center 0) with n terms and remainder bounds.

Calculator

Maclaurin Series Inputs (c = 0)

Point (x)

Terms (n)

Function (center fixed at 0)

Results update live as you type

Status:M = 0.4794270833true = 0.4794255386 · err 0.00000154472913

Maclaurin Series Summary

Maclaurin sum (c=0)0.4794270833

True value0.4794255386

Setupsin · x=0.5, n=7

Term list

0, 0.5, 0, -0.02083333333, 0, 0.0002604166667, 0

c is fixed at 0 on this page — use Taylor for other centers. Trig uses radians.

Results

Optimal

Maclaurin Sum M(x)

0.4794270833

c=0 polynomial

True f(x)

0.4794255386

Direct evaluation

Actual Error

0.00000154472913

|true − approx|

Remainder Bound

0.000001550099206

Worst-case ceiling

Function

sin

Active family

Terms n

Depth

Zero-centered coefficients with Lagrange fencing; ln domain guarded at x > −1.

Visualizations

Maclaurin term contributions

0a1

0.5a2

0.5a3

0.5a4

0.5a5

0.5a6

0.5a7

Convergence toward f(x)

Terms used

n / 507

Approximation vs true value

approx0.5

Formula

Taylor fixed at zero: derivatives at the origin become universal coefficients for each family.

M(x) = Σ f^(k)(0)/k! · x^k

Alternative Form

Remainder: |R| ≤ M·|x|^(n+1)/(n+1)!

Variables

Symbol	Name	Unit	Description
f	Function		exp / sin / cos / ln(1+x)
x	Point	number	Where to approximate (radians trig)
n	Terms used	count	Polynomial depth (1..50)
R	Remainder bound	number	Worst-case error fence

Explanation

Zero-centered powers are the simplest basis: x^k needs no displacement, and classic coefficients are memorizable.

Derivation

Taylor at c=0 with Lagrange remainder. Coefficients: exp 1/k!; sin/cos alternating odd/even; ln(1+x) (−1)^(k+1)/k.

Step-by-Step Calculation

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

Step 1

Read the function, point, and depth (c = 0 fixed).

Inputs: f, x, n; c = 0

func:sin

x:0.5

n:7

Result: sin at x=0.5, n=7

Step 2

Look up zero-centered coefficients.

f^(k)(0)/k!

Intermediate: 0, 0.5, 0, -0.02083333333, 0, 0.0002604166667, 0

Result: 7 terms formed

Step 3

Weight by bare powers of x.

coef(k)·x^k

Intermediate: x = 0.5

Result: M = 0.4794270833

Step 4

Compare with the true value.

|f(x) − M|

Intermediate: 0.4794255386 vs 0.4794270833

Result: error = 0.00000154472913

Step 5

Fence the error.

M|x|^n/n! scale

Intermediate: ≤ 0.000001550099206

Result: bound = 0.000001550099206

Step 6

Certify digits.

error ≤ bound ?

Intermediate: error 0.00000154472913 vs bound 0.000001550099206

Result: Bound holds

Final Answer

0.4794270833

Worked Examples

See how the calculator works with different scenarios.

Scenario

exp at x=1, n=7.

Inputs

func:exp

x:1

n:7

Calculation

M ≈ 2.71806

Result

M ≈ 2.71806

Interpretation

Near-e in seven adds.

Understanding Your Result

Your Result: Maclaurin sum at c = 0 with error fencing — see the cards.

Typical range: n caps at 50; ln needs x > −1; trig in radians.

What This Means

The zero-centered polynomial value plus its certified worst-case error.

Why It Matters

Embedded math libraries ship these exact polynomials — this page shows what your chip evaluates.

Professional Interpretation

Because c = 0 always, distance is just |x| — convergence pace reads off immediately.

Recommendations

•Use small |x| for speed
•Grow n with |x|
•Respect ln domain
•Read bound as ceiling
•Compare with true value

Educational Content

Introduction

No center to choose — zero does the anchoring. Pick the function, name x, depth n, done.

Concept Explanation

Maclaurin = Taylor at the origin: canonical coefficients, displacement-free powers.

Applications

▸Chip-level exp/log kernels

▸Small-angle physics

▸Audio oscillator tables

▸Probability tail sums

▸Graphics easing curves

▸Compound-growth sketches

Benefits

•Zero-center simplicity
•Four families
•True-value check
•Bound included
•Term trace

Limitations

•c fixed at 0
•Four functions
•n <= 50

Best Practices

•Keep |x| modest
•Raise n with distance
•Check domain first
•Verify n=1 tangent
•Quote bound with digits

Common Misconceptions

Myth: Maclaurin and Taylor are unrelated.

Truth: Maclaurin IS Taylor with c = 0.

Myth: The bound equals the error.

Truth: Ceiling, not value — usually pessimistic.

Myth: Any x works for ln.

Truth: Needs x > −1 (radius 1 at c=0).

Myth: Trig series take degrees.

Truth: Radians only.

Practical Tips

•Keep |x| small for fast convergence
•Grow n with distance from 0
•Respect x > −1 for ln(1+x)
•Use radians for sin/cos
•Read the bound before quoting digits
•Check the n=1 linearization
•Compare poly vs true directly

Common Mistakes

Recentering a Maclaurin sum

c is fixed 0 here — use the Taylor page for c ≠ 0.

Degrees in trig inputs

Convert to radians first.

ln at x ≤ −1

Outside domain — flagged, not computed.

Too few terms far out

Raise n; error grows like |x|^n/n!.

Quoting past the bound

Uncertified digits are decoration.

Assumptions

This calculator makes the following assumptions to provide estimates:

Center zero: c = 0 always.

Real x: Radians for trig.

Truncation n: n terms ≤ 50.

Estimated M: Per-family bound constant.

Local only: Browser doubles.

Limitations

Be aware of these limitations when using this calculator:

Fixed center: c ≠ 0 needs Taylor page.

Four families: Custom f unsupported.

Radius limits: ln converges for |x| < 1 (plus x=1).

Cancellation: Far alternating sums round.

Frequently Asked Questions

Answers to common questions about this calculator.

Maclaurin vs Taylor?

Maclaurin fixes c=0; Taylor allows any center. Same formula otherwise.

e^x Maclaurin?

1 + x + x²/2! + x³/3! + ... — every coefficient 1/k!.

sin x Maclaurin?

x − x³/6 + x⁵/120 − ...

cos x Maclaurin?

1 − x²/2 + x⁴/24 − ...

ln(1+x) Maclaurin?

x − x²/2 + x³/3 − ... for −1 < x ≤ 1.

Example e^0.5, 6 terms?

≈ 1.6487 vs true 1.64872 — five-decimal agreement.

Why is c=0 convenient?

Powers are bare x^k and coefficients are famous constants.

When does ln fail?

x ≤ −1 (singularity) and |x| > 1 (radius).

How many terms?

Until |x|^n/n! beats your tolerance.

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.

Maclaurin series: Taylor series at c = 0.
Coefficients: f^(k)(0)/k! per family.
Remainder bound: Worst-case truncation fence.
Radius 1 (ln): ln(1+x) Maclaurin converges |x|<1.
Radians: Native trig-series angle unit.
Linearization: n=1 (tangent) approximation.
Alternation: Sign flips in sin/cos/ln.
True value: Direct Math.* result.
Depth n: Terms kept.

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.