Calculus II Final 9.3-9.6, 10.1-10.2 Flashcards
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
r=lim(ak+1/ak)
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)|
