AP Calc Series Concept Test Review

What is a sequence?

a list of numbers, that often has a rule to define it

What is a series?

the sum of a list of numbers

What does it mean when a equation is bounded?

It doesn’t go past a certain point (can only be bounded in one direction x or y(??))

What does it mean when an equation is convergent?

as n → infinity, a_n or s_n approaches a number

What does it mean if an equation is divergent?

As n → infinity, a_n or s_n doesn’t approach a number

What does it mean if an equation is monotonic?

strictly going up or down

If a_n is bounded and monotonic, then a_n …

converges

If a_n is convergent, then it is

bounded

What is a geometric series?

(x)ⁿ = a_n

What value does a geometric series need to be multiplied by to converge?

-1 < x < 1

What is the equation for a finite geometric sum?

a (1-xⁿ) / (1-x)

What is the equation for an infinite geometric sum?

a₁ / (1-r)

How does the nth term test for divergence work?

For S a_n if a_n doesn’t approach 0, then S a_n diverges

Changing the finite number of terms ___ change whether the series converges or diverges

cannot

What is a harmonic series?\n\n

A harmonic series is the infinite series formed by the reciprocals of the positive integers, expressed as \sum_{n=1}^{\infty} \frac{1}{n} .

If the S a_n and S b_n both converge

S a_n + b_n converges to Sa_n + Sb_n

S ka_n converges to k S a_n

If S a_n diverges, then S ka_n for k (where k is not 0) …

diverges

If the limit as n → infinity a_n doesn’t equal 0, S a_n …

diverges

What is the integral test?

For a P-Series test what value of P makes the values converge?

p>1

What is the P-Series Test?

For a P-Series test what value of P makes the values diverge?

p <= 1

What is the comparison test?

What is the limit comparison test?

What is the layout for the answer to a Sum test?

Sum (converges/diverges) by (type of test) with (details of equation if comparison).

What is the ratio test? (Include values in description)

What is the alternating series test?

What does it mean to be conditionally convergent?

What does it mean to be absolutely convergent?

What is the interval of convergence?

The symmetric interval around “a”(the center of a power series), where a functions values converge

How do you find the radius of convergence?

Use ratio test, find it’s limit and the radius is the reciprocal of the limit.

If the limit is 0, R is infinite

If the limit is infinite, R is 0

If the limit is finite, R = 1/L

What is the ratio of convergence?

The distance from “a” to the boundary in the interval of convergence.

What is a power series?

How is a Taylor Polynomial different from a tangent line?

A Taylor Polynomial will not only follow the slope of the line at said point, but will follow its curvature

What is the difference between a Taylor and Maclaurin Polynomial?

A Taylor Polynomial is centered at any point, where a Maclaurin is centered at 0

What are the 4 Maclaurin terms for y= sinx?

x- x³/3! + x⁵/5! - x⁷/7!

What are the 4 Maclaurin terms for y= cosx?

1- x²/2 + x⁴/4! - x⁶/6!

What are the 4 Maclaurin terms for y= e^x?

1+x+ x²/2 + x³/3!

What are the 4 Maclaurin terms for y= 1/(1+x)?

1-x+x² -x³

If the Taylor Series has a factorial what test should you use?

Ratio Test

What is a binomial series?

A binomial series is the expansion of expressions in the form (1 + x)^k, where k is any real number