cheatsheet - IMPROPER INTEGRALS p if p > 1 converges 1 dx/x...

IMPROPER INTEGRALS
1
∫ ∞ dx/x p if p > 1  converges
if p ≤1  diverges
Direct Comparison:
g(x) ≥ f(x)  if ∫g(x)dx C, so does ∫f(x)dx
 if ∫f(x)dx D, so does ∫g(x)dx
Limit Comparison:
If lim g(x) = L then
a
∫ ∞ f(x)dx and
a
∫ ∞ g(x)dx
x

∞ f(x)
both C or D
INFINITE SERIES
1) Check if lim a
n
= 0 ; if NO, series is D
n  ∞
∞
2) Geometric ? Is it of the form: ∑ ar (n-1) = a/(1-r)
n=1
 if |r| < 1, then converges to (1-r)
 if |r| ≥ 1, then diverges
∞
3) Telescoping ? Is if of the form: ∑ (a
n
– a
n+1
)
n=1
if a
n 
0, then sum = a
1
-try rewriting with partial fractions
∞
4) Is it of the form ∑ 1/n p
n=1
if p > 1  C; if p ≤ 1  D
5) Direct Comparison Test :
∞
If a
n
> 0 for all n AND a
n
≤ c
n
and ∑ c
n
is C,
∞
n=1
then ∑ a
n
is C
n=1
∞ ∞
 if a
n
≥ d
n
and ∑ d
n
is D , then ∑ a
n
is D
n=1 n=1
6) Limit Comparison Test :
(put what you are comparing w/ on the bottom )
If lim a
n
/b
n
= c > 0, then ∑a
n
& ∑b
n
both C or D
n  ∞
= 0 then C of ∑b
n
guarantees C of ∑a
n
= ∞ then D of ∑b
n
guarantees D of ∑a
n
7) Ratio Test : (good w/ n!’s)
If lim a
n+1
/a
n
<1, then C
n  ∞ >1, then D
=1, inconclusive
9) Alternating Series Theorem
(check absolute convergence first, then use i) ii) iii)
to check for conditional convergence)
∞
∑(-1) n a
n
converges if:
n=1 i) a
n
> 0 for n>N
ii) a
n+1
≤ a
n
for n>N
iii) lim a
n
= 0
n  ∞
-interval of C is when the end points are checked
(if converges, then is also equal to the endpoint)
-absolute interval of C is the interval given by the
ratio test used to find the limit
-radius is what the limit goes to
-conditionally convergent where x makes CC
 If ∑ a
n
is C, then series is absolutely convergent
 If ∑ (-1) n+1 a
n
is C but ∑a
n
is D then ∑ (-1) n+1 a
n
is
conditionally convergent .
∞
10) Power Series : power series about b: ∑C
n
(x-b) n
1)Ratio Test it! n=0
2) get radius of convergence
3)Test end points (plug x’s back into original series & see if
series converges with a test):
 if C, then (=) in interval of convergence and converges
conditionally at that x
 interval of absolute conv. Stays same except if end point x
causes absolute convergence (in alt. series)
∞
11) Taylor Series : for f(x) about x=a is ∑ f (n) (a) (x-a) n
n=0 n!