Using and Manipulating Series

Building new series from known ones.

12 problems · about 22 minutes · LIM-8.F, LIM-8.G

What this lesson teaches

The student manipulates known Maclaurin series and bounds the error of a Taylor polynomial.

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.

Straightforward practice. Get the method working first.

5 problems

Substituting 5x into the eˣ series. What is the coefficient of x?
Answer 5
Why 5.
Why is the Lagrange remainder a bound rather than an exact value?
Answer The exact form needs an unknown point, so the derivative is bounded instead.
Why The unknown point forces a bound.
In the series for eˣ, what is the constant term?
Answer 1
Why e to the zero.
In the series for eˣ, the x² term is x² ÷ k. What is k?
Answer 2
Why 2 factorial.
In the series for eˣ, the x³ term is x³ ÷ k. What is k?
Answer 6
Why 3 factorial.

The same ideas with more to keep track of.

3 problems

Substituting 3x into the eˣ series. Coefficient of x?
Answer 3
Why x becomes 3x.
May a power series be differentiated term by term inside its radius? 1 yes, 0 no.
Answer 1
Why Within the radius it is legal.
The sine series has only which powers? 1 odd, 2 even.
Answer 1
Why Sine is an odd function.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Which operations build a new series from a known one?
Answer Substituting an expression for x; Differentiating term by term; Integrating term by term
Why Every operation still respects the radius.
The cosine series has only which powers? 1 odd, 2 even.
Answer 2
Why Cosine is an even function.
The Substitution: In the series for eˣ, the x term has coefficient 1. After substituting 2x, what is the coefficient of x?
Answer 2
Why 2.
The Known Series: How many Maclaurin series are named as worth knowing cold?
Answer 4
Why 4.