Common Maclaurin series
Use standard power series for exponential, sine, and cosine.
Common Maclaurin series is one of 6 series formulas in the calculus section of this library, and it is used at ap · university level.
Why common maclaurin series works
Each of the three is what the Taylor recipe produces when the centre is 0. The exponential reproduces its own derivative, so every coefficient is one over a factorial, while sine and cosine cycle through four derivatives, and that cycle is what makes the signs alternate and half the powers drop out.
What each symbol means
$n$ indexes terms and factorials determine coefficients.
Common Maclaurin series: when it holds
These three series converge for every real or complex $x$.
When it stops applying
All three converge for every real input, so there is no value where the mathematics breaks. On a computer, though, a large input makes enormous terms cancel against each other, and the answer loses most of its accurate digits.
Common Maclaurin series: a worked example
Near $0$, $\sin x\approx x-x^3/6$.
The mistake to avoid
What people do: Writing the sine series with every power of the variable rather than only the odd ones.
Why it goes wrong: Sine is an odd function, so all of its even-power coefficients are zero, starting with the constant term, since the sine of 0 is 0.
Do this instead: Keep odd powers for sine and even powers for cosine, alternating the signs in both.
Common Maclaurin series: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $n$ indexes terms and factorials determine coefficients.
- Check the conditions before substituting. These three series converge for every real or complex $x$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Series
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where common maclaurin series comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Sequences and Series — the lesson behind this formula: recognize arithmetic, geometric, and recursive patterns.
- Taylor Series Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about common maclaurin series
Why does the exponential series have no alternating signs?
Its derivatives at the centre are all equal to 1, never negative, so every coefficient comes out positive.
How accurate is keeping only two terms of the sine series?
At an input of 0.5 the two-term estimate gives 0.4791667 against the true 0.4794255, so the error is under 0.0003.
How are the three series connected?
Feeding an imaginary input into the exponential series splits it into the cosine series plus the sine series, which is Euler's famous formula.
Can I build other series from these?
Yes. Substituting, differentiating or multiplying produces new ones, so the series for e to the minus x squared comes free from the exponential one.