Four Maclaurin series are worth knowing cold: those for eˣ, sine, cosine, and 1 ÷ (1 − x).
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Substitution
Replacing x with something else in a known series gives a new one instantly. The series for e^(x²) needs no new derivatives.
Differentiate and integrate termwise
Within the radius of convergence, a power series may be differentiated or integrated term by term.
The geometric series is a generator
Substituting and integrating the series for 1 ÷ (1 − x) produces the series for the logarithm and the arctangent.
The Lagrange error bound
Truncating a Taylor series leaves an error bounded by the next term with the derivative replaced by its maximum on the interval.
Why a bound rather than a value
The exact error needs an unknown point c. Bounding the derivative gives a usable guarantee instead.
Substituting into a known series
The series for e^(x²) comes from the series for eˣ with x² substituted. This is far faster than differentiating repeatedly and is the intended method in most exam questions.
Differentiating and integrating termwise
Within its interval of convergence, a power series may be differentiated or integrated term by term, and the radius is unchanged. This is how the series for arctan and ln(1 + x) are obtained.
Multiplying and adding
Series can be added termwise and multiplied like polynomials, keeping terms up to the degree required. For a few terms this is quick, and it avoids computing high derivatives of a product.
Series evaluate hard limits
Replacing a function by its series often resolves an indeterminate form immediately, since the lowest-order terms dominate near the centre. It is frequently faster than repeated L'Hôpital.
Step 2: Try It Yourself
Tap and try it out.
- Point(0, 1)
Step 3: Watch an Example
One step at a time.
Watch Kofi Build a Series by Substitution
Kofi needs the Maclaurin series for e^(2x).
- Step 1
He starts from the known series for eˣ: 1 + x + x²/2 + x³/6 and so on.
Step 4: Your Turn
Practice makes it stick.
The Substitution
Problem 1 of 2
In the series for eˣ, the x term has coefficient 1. After substituting 2x, what is the coefficient of x?
The Known Series
Problem 2 of 2
How many Maclaurin series are named as worth knowing cold?
Manipulate a Series
1 of 8
In the series for eˣ, what is the constant term?
2 of 8
In the series for eˣ, the x² term is x² ÷ k. What is k?
3 of 8
In the series for eˣ, the x³ term is x³ ÷ k. What is k?
4 of 8
Substituting 3x into the eˣ series. Coefficient of x?
5 of 8
May a power series be differentiated term by term inside its radius? 1 yes, 0 no.
6 of 8
The sine series has only which powers? 1 odd, 2 even.
7 of 8
Which operations build a new series from a known one?
8 of 8
The cosine series has only which powers? 1 odd, 2 even.
Step 5: Quick Check
Show what you know.
Question 1 of 2
Substituting 5x into the eˣ series. What is the coefficient of x?
Question 2 of 2
Why is the Lagrange remainder a bound rather than an exact value?
What You Learned
- Four Maclaurin series generate most others by substitution.
- Power series may be differentiated and integrated term by term inside the radius.
- The Lagrange bound uses the maximum derivative because the exact point is unknown.