Using and Manipulating Series

Building new series from known ones.

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

NameDate

Figure — use these to answer the problems

A Taylor polynomial hugs the curve near the centre and drifts away from it further out.

y = 1 · 2^x + 0

Point on the curve0

Figure 1

Straightforward practice. Get the method working first.

5 problems

Substituting 5x into the eˣ series. What is the coefficient of x?
Answer
2
Why is the Lagrange remainder a bound rather than an exact value?
- The exact form needs an unknown point, so the derivative is bounded instead.
- Because computing it exactly is tedious.
In the series for eˣ, what is the constant term?
Answer
In the series for eˣ, the x² term is x² ÷ k. What is k?
Answer
In the series for eˣ, the x³ term is x³ ÷ k. What is k?
Answer

The same ideas with more to keep track of.

3 problems

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

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

4 problems

9
Which operations build a new series from a known one?
- Substituting an expression for x
- Differentiating term by term
- Integrating term by term
- Ignoring the radius of convergence
- Tick every box that applies.
The cosine series has only which powers? 1 odd, 2 even.
Answer
The Substitution
In the series for eˣ, the x term has coefficient 1. After substituting 2x, what is the coefficient of x?
Answer
The Known Series
How many Maclaurin series are named as worth knowing cold?
Answer