zuai-logo
zuai-logo
  2. AP Calculus
FlashcardFlashcard
Study GuideStudy GuideQuestion BankQuestion BankGlossaryGlossary

What is the general form of a power series?

∑n=0∞an(x−r)n\sum^{\infty}_{n=0}a_n(x-r)^n∑n=0∞​an​(x−r)n

Flip to see [answer/question]
Flip to see [answer/question]
Revise later
SpaceTo flip
If confident

All Flashcards

What is the general form of a power series?

∑n=0∞an(x−r)n\sum^{\infty}_{n=0}a_n(x-r)^n∑n=0∞​an​(x−r)n

Write the formula for the Ratio Test.

L=lim⁡n→∞∣an+1an∣L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|L=limn→∞​​an​an+1​​​

What condition on L in the Ratio Test implies convergence?

L<1L < 1L<1

What condition on L in the Ratio Test implies divergence?

L>1L > 1L>1

What does the Ratio Test tell you about the convergence of a series?

For a series ∑an\sum a_n∑an​, let L=∣limlimitsn→∞an+1an∣L=|limlimits_{n→ \infty}\frac{a_{n+1}}{a_n}|L=∣limlimitsn→∞​an​an+1​​∣. If L < 1, the series converges. If L > 1, the series diverges. If L = 1, the test is indeterminate.

Explain the purpose of the Ratio Test in determining convergence of a power series.

The Ratio Test helps determine the radius of convergence by finding the values of x for which the limit L < 1.

Why is it necessary to test the endpoints of the interval obtained from the Ratio Test?

The Ratio Test only provides an open interval. Endpoints must be tested individually to determine if they are included in the interval of convergence.

What does the radius of convergence tell you?

The radius of convergence tells you how far from the center the power series converges.

How do power series relate to functions?

Power series can represent functions over an appropriate interval. They are approximations of functions.

What happens when L = 1 in the Ratio Test?

The Ratio Test is inconclusive. Another test must be used to determine convergence or divergence.