Calculus II Final 9.3-9.6, 10.1-10.2 Flashcards Preview

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Flashcards in Calculus II Final 9.3-9.6, 10.1-10.2 Deck (16):

Geometric Series

series of the form r^n
if |r|


Telescopic Series of the form 1/(n(n+1)

= ((1/n)-(1/n-1)) converges to 1-1(n+1)


Telescopic Series of the form 1/(a^n)-1/(a^(n+1))

converges to 1-1/(a^(n+1))


Divergence test

if the series convenes then the lim of the sequence=0
if the lim of the sequence is not=0 then the series diverges


Integral test

if f is a continuous, positive, and decreasing function;
set an=f(n)
then the series an converges if the integral of f(n) is less than infinity


Ratio test

ak is a series such that ak>0
if 01 then ak diverges
if r=1 then the ratio test is inconclusive


Absolute and conditional convergence

1)a series ak is absolutely convergent if |ak| converges
2) if series ak converges but |ak| diverges then ak converges conditionally


limit comparison test

find a bk to compare ak with
take the lim of ak/bk
if the lim= to a # then the series diverges


Comparison test

Ak is the series your evaluating, bk is the series you pick that is similar but easier to evaluate
if ak is less than bk and bk converges, then ak converges
if bk is less than ak and bk diverges, then ak diverges


Alternating series test

A series of the form (-1)^(k+1)*ak
if the limit as k->infinity of ak=0
then the series converges


What is a MacLaurin series?

A taylor series in which a=0


Taylor Polinomial

Pn(x)=f(a)+f'(a)*(x-a)+(f"(a)/2!)*(x-a)^2.....+(f^n(a)/n!)*(x-a)^n where f(x) is centered at a


Taylor series

series (f^k(a)/k!)*(x-a)^k


Power series

series Ck(x-a)^k where Ck are the coefficients of the power series and the function is centered at a


How to find the radius of convergence(R)

R=lim as k->infinity of 1/|(Ck+1)/(Ck)|


process to find where a series converges and diverges on the entire number line

1. find the radius of converges: inside this the series converges, outside it diverges
2. evaluate the series at the end points of R