AP Calc BC Unit 10

Geometric series test

absolute value of r is greater than or equal to 1 then it diverges

absolute value of r is less than 1 then it converges

Hint: this is the only series that we can find out what a series converges to.

nth term test for divergence

if the limit as n goes to infinity of an does not equal 0, then an diverges.

Hint: always try this test first.

comparison test for series

if a series, bn, is greater than a series, an, and bn converges, then an must converge. if an is greater than bn, and bn diverges, then an must converge. You can’t trap a divergent with a convergent, and you can’t trap a convergent with a divergent.

harmonic series

1/n diverges

integral test

f has to be continuous, positive, and decreasing.

whatever the integral goes to, the series must go to.

it is convergent if it goes to a number.

it is divergent if it goes to ± infinity.

p-series test

the p series 1/n^p converges if p>1

it diverges if p is < or = to 1.

limit comparison test

an/bn=L. L must be 0<L<infinity, then both series converge or diverge.

alternating series test

a series whose terms alternate positive and negative (-1)^n or (-1)^n-1 or (-1)^n+1.

if every an is positive and each term is less than or = to the previous term and if the lim = 0 then the series is convergent.

what does “the sum of an infinite series” indicate

geometric series

What is error?

The sum of the first term left off and the rest of the terms.

S= Sn + Rn (Rn is the actual sum minus the nth partial sum)

Ratio test

if lim [an+1/an] < 1, then the series an is absolutely convergent (and therefore convergent)

if lim [an+1/an] > 1 then the series an is divergent

if lim [an+1/an] =1, the ratio test is inconclusive.

this test is used a lot with factorials and (something)^n. Dividing will simplify many things to 1.

absolutely convergent

a series is absolutely convergent if [an] is convergent

conditionally convergent

if an is convergent but [an] is not convergent

taylor series formula

Tn(x)= sigma notation (f^k(a) (x-a)^k)/k!

e^x Taylor series

1 + x + x²/2! + x³/3! + x^4/4! + … + x^n/n!

sinx Taylor series

x - x³/3! + x^5/5! - x^7/7! + … alternating terms of odd degree

cosx taylor series

1 - x²/2! + x^4/4! - x^6/6! + … alternating terms of even degree

1/1-x Taylor series

1 + x + x² + x³ + x^4 + ….

what happens when a geometric series converges?

it converges when [r] is < 1 to a1/1-r

la grange error bound

f(x) = Tn(x) + Rn(x)

[Rn(x)] < or = max [f^n+1(z)]/(n+1)! [x-a]^n+1

Alternating series error bound

Rn = [S-Sn] < or = an+1