AtlasCalcsUtility Calculation Suite
2026-09-29 10 min

Taylor Series Calculator

Approximate e^x, sin x, cos x, ln(1+x) by Taylor polynomials around c with n terms plus a remainder bound.

Calculator

Taylor Series Inputs

Center (c)
Point (x)
Terms (n)
Function
Results update live as you type
Status:T = 2.718253968true = 2.718281828 · err 0.00002786020508

Taylor Series Summary

Taylor sum2.718253968
True value2.718281828
Setupexp · c=0, x=1, n=8

Term list

1, 1, 0.5, 0.1666666667, 0.04166666667, 0.008333333333, 0.001388888889, 0.0001984126984

Bars show each term\'s signed contribution; use radians for trig.

Results

Optimal

Taylor Sum T(x)

2.718253968

n-term polynomial

True f(x)

2.718281828

Direct evaluation

Actual Error

0.00002786020508

|true − approx|

Remainder Bound

0.0001832603199

Worst-case ceiling

Function

exp

Active family

Terms n

8

Terms used

Derivatives at c supply coefficients; the Lagrange bound fences the truncation error.

Visualizations

Taylor term contributions

1a1
2a2
2.5a3
2.7a4
2.7a5
2.7a6
2.7a7
2.7a8

Convergence toward f(x)

Terms used

n / 508

Approximation vs true value

approx2.7

Formula

Derivatives at the center become coefficients; the next-derivative maximum bounds what was chopped off.

T(x) = Σ f^(k)(c)/k! · (x−c)^k

Alternative Form

Remainder: |R| ≤ M·|x−c|^(n+1)/(n+1)!

Variables

SymbolNameUnitDescription
fFunctionexp / sin / cos / ln(1+x)
cCenternumberExpansion point
nTerms usedcountHow many series terms (1..50)
RRemainder boundnumberWorst-case truncation error

Explanation

Smooth functions are locally polynomial: match derivatives at c and the neighborhood follows.

Derivation

Taylor with Lagrange remainder: f(x) = T(n) + f^(n+1)(ξ)(x−c)^(n+1)/(n+1)!. Bound |f^(n+1)| by M on the span.

Step-by-Step Calculation

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

Step 1

Read function, center, point, and depth.

Inputs: f, c, x, n
func:exp
c:0
x:1
n:8
Result: exp at c=0, x=1, n=8
Step 2

Differentiate at the center for coefficients.

f^(k)(c)/k!
Intermediate: 1, 1, 0.5, 0.1666666667, 0.04166666667, 0.008333333333, 0.001388888889, 0.0001984126984
Result: 8 terms formed
Step 3

Weight by displacement powers.

term(k)·(x−c)^k
Intermediate: h = 1
Result: T = 2.718253968
Step 4

Compare against the true value.

|f(x) − T|
Intermediate: 2.718281828 vs 2.718253968
Result: error = 0.00002786020508
Step 5

Fence the error with Lagrange.

M|h|^(n+1)/(n+1)! scale
Intermediate: ≤ 0.0001832603199
Result: bound = 0.0001832603199
Step 6

Certify quoted digits.

error ≤ bound ?
Intermediate: error 0.00002786020508 vs bound 0.0001832603199
Result: Bound holds
Final Answer
2.718253968

Worked Examples

See how the calculator works with different scenarios.

Scenario

exp, c=0, x=1, n=8.

Inputs

func:exp
c:0
x:1
n:8

Calculation

T ≈ 2.718254, bound ≈ 1.8e−4

Result

T ≈ 2.718254

Interpretation

Five-decimal e from a polynomial.

Understanding Your Result

Your Result: Taylor sum and its error bound for your f, c, x, n — see the cards.

Typical range: n caps at 50; ln(1+x) needs x > −1. M estimated as e^max for exp, 1 for trig, n!-style for log.

What This Means

The n-term polynomial value plus a guaranteed worst-case distance from the true function.

Why It Matters

Calculators and physics engines evaluate sin/exp this way — bounds certify the digits.

Professional Interpretation

The bound is honest worst-case: actual error is usually far smaller, never larger (within M's validity).

Recommendations

  • •Center near x for speed
  • •Add terms when far
  • •Respect ln domain x>−1
  • •Read bound as worst case
  • •Verify n=1 is the tangent

Educational Content

Introduction

Replace curves with polynomials around your center — and know exactly how much was lost.

Concept Explanation

Derivative-matching plus a fenced-off remainder: approximation with a certificate.

Applications

▸Calculator sin/exp kernels
▸Pendulum small-angle tweaks
▸Control-system linearization
▸Option-greek expansions
▸Antenna pattern sums
▸Log-domain compressions

Benefits

  • •Four built-in functions
  • •Any center c
  • •Remainder bound
  • •Term trace
  • •True-value comparison

Limitations

  • •Four functions only
  • •Bound uses estimated M
  • •n <= 50

Best Practices

  • •Keep c close to x
  • •Grow n with distance
  • •Check ln domain
  • •Compare true vs poly
  • •Read bound before quoting digits

Common Misconceptions

Myth: More terms always add accuracy.
Truth: Inside the radius yes; ln far from 1 stalls — recenter instead.
Myth: The bound is the error.
Truth: It is the ceiling; true error is typically much smaller.
Myth: Center is cosmetic.
Truth: It sets every coefficient — c near x converges fastest.
Myth: Taylor equals the function everywhere.
Truth: Only within the convergence interval (all reals here except ln).

Practical Tips

  • •Center close to x converges fastest
  • •Grow n with |x−c|
  • •Respect x > −1 for ln(1+x)
  • •Read the bound as worst case
  • •Check n = 1 is the tangent line
  • •Compare poly vs true value
  • •Add terms before moving centers

Common Mistakes

Centering ln at c ≤ −1
ln(1+x) expands around c > −1 — recenter inside the domain.
Using degrees in trig series
Series need radians; convert first.
Quoting full precision past the bound
Digits beyond the bound are not certified.
Too few terms far out
Raise n or slide c toward x.
Ignoring alternating cancellation
Far trig sums round — prefer nearer centers.

Assumptions

This calculator makes the following assumptions to provide estimates:

Smooth functions: Derivatives exist to needed order.
Real inputs: x, c real; radians for trig.
Truncated poly: n terms, n ≤ 50.
Estimated M: Bound constant per family.
Local only: Browser doubles.

Limitations

Be aware of these limitations when using this calculator:

Four families: Arbitrary f needs manual coefficients.
Bound estimates: M heuristic, not sup-norm proof.
Radius limits: ln valid for x > −1.
Cancellation: Long alternating sums round.

Frequently Asked Questions

Answers to common questions about this calculator.

What is e^1 to 8 terms?
c=0,x=1,n=8 → ≈ 2.718254 (true 2.718282, bound ≈ 1.8e-4).
Sin series?
x − x³/6 + x⁵/120 − ... (odd powers, alternating).
Cos series?
1 − x²/2 + x⁴/24 − ... (even powers, alternating).
ln(1+x) series?
x − x²/2 + x³/3 − ... for x > −1 (centered 0).
What is the remainder?
The chopped tail, bounded by M|x−c|^(n+1)/(n+1)!.
Why center near x?
Powers of (x−c) shrink then — fewer terms, tighter bound.
Degrees or radians?
Radians — series are radian-native.
How many terms?
Until the bound beats your tolerance; distance sets the pace.
Maclaurin difference?
Just c = 0 framing — see the sibling page.
Stored?
No.

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Related Guides

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Glossary

Technical terms used in mortgage calculations explained in plain language.

Taylor polynomial
Truncated derivative-matched sum.
Center
c where derivatives match.
Remainder
f(x) − T(n); truncation error.
Lagrange bound
M|x−c|^(n+1)/(n+1)! ceiling.
Radius of convergence
Range where series → f.
Maclaurin
Taylor at c = 0.
Alternating series
Sign-flipping terms (sin/cos).
Derivative match
Equal derivatives at c to order n.
True value
Direct Math.* evaluation.

Authoritative References

Trusted sources and standards referenced by this calculator.

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

License:

Free to use for educational and professional purposes.