height and slope
🔎CALCULUS II STUDY GUIDE
Power Series and Taylor Series
Find intervals of convergence and build polynomial approximations with Taylor and Maclaurin series.
8 key ideas8 flashcards5 practice questions
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More derivative matches give a closer local polynomial.
adds curvature
matches derivatives through n
Key ideas to know
Start with the relationships between ideas. Then close the notes and explain each one from memory.
- 01
A power series is a series of coefficients times powers of x−c.
- 02
Its radius of convergence gives the distance from center c to each endpoint.
- 03
The ratio test commonly finds the radius, but endpoints need separate tests.
- 04
A power series may be differentiated or integrated term by term inside its interval of convergence.
- 05
A Taylor polynomial matches function derivatives at its center.
- 06
A Maclaurin series is a Taylor series centered at zero.
- 07
Known series can be shifted, scaled, differentiated, or integrated to produce new ones.
- 08
The Taylor remainder measures the gap between a function and its polynomial approximation.
PART OF THE CALCULUS II GUIDE
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Open the full Calculus II guide →Two ideas worth correcting now
NOT QUITE
→The radius test includes endpoints
USE THIS INSTEAD
Endpoints always need their own tests.
NOT QUITE
→Every infinitely differentiable function equals its Taylor series
USE THIS INSTEAD
The Taylor remainder must tend to zero for equality on an interval.
Flashcards
Answer before opening each card. The effort to retrieve is part of the learning.
1What is the center of Σaₙ(x−c)ⁿ?Show answer ＋
c.
2What does the radius R describe?Show answer ＋
Convergence for |x−c|<R and divergence for |x−c|>R.
3How are endpoints tested?Show answer ＋
Substitute each endpoint into the original series and test separately.
4What makes a Maclaurin series special?Show answer ＋
Its center is zero.
5What coefficient multiplies (x−c)ⁿ in a Taylor series?Show answer ＋
f⁽ⁿ⁾(c)/n!.
6What is the geometric power series?Show answer ＋
1/(1−x) = Σxⁿ for |x|<1.
7Where is termwise differentiation valid?Show answer ＋
Inside the radius of convergence.
8What does a remainder bound estimate?Show answer ＋
The approximation error after a finite Taylor polynomial.
Explain it in your own words
Use the answer as a check after you have written or spoken your response.
01Why can endpoints behave differently?
The ratio test gives the open radius, while endpoint substitution can produce two different numerical series.
02Build the first three nonzero terms of eˣ at zero.
1 + x + x²/2.
03How can a series for 1/(1+x²) be obtained?
Replace x by −x² in the geometric series, giving Σ(−1)ⁿx²ⁿ for |x|<1.
04Why do derivative matches matter?
They make the polynomial agree with local function behavior through the chosen order.
05How is the needed polynomial degree selected?
Use a remainder bound and raise the degree until the stated error tolerance is met.
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