LBS 119 – EXAM #1 – REVIEW SHEET
KNOW ALL OF THE FOLLOWING TESTS!
A list WILL NOT
be given to you for the exam!
1.
GEOMETRIC series CONVERGE to a sum of
1
a
r

if
1
r
<
.
(Otherwise they diverge.)
2.
TELESCOPING series – use partial fractions and then write out the terms of the nth partial sum and show what
cancels and then take the limit as n goes to infinity.
3.
HARMONIC series DIVERGES.
(But the alternating
harmonic converges.)
4.
N
th
TERM TEST:
If
lim
0
n
n
a
r
°
r
, then the series DIVERGES.
REMEMBER:
You cannot tell what happens if the limit is zero…
This is only a test for DIVERGING series.
5.
INTEGRAL TEST:
The series must be positive, continuous and decreasing before this test can be used.
The test
says that BOTH the integral and the series CONVERGE or BOTH DIVERGE.
6.
PSERIES:
pseries CONVERGE when
1
p
.
(Otherwise diverge.)
(But, ALL alternating
pseries CONVERGE)
7.
DIRECT COMPARISON TEST.
To compare your unknown series to one you know about, place the nth term
formulas side by side and place a less than or greater than sign between them.
This should be immediately obvious
in order for you to use the test!
If it is not obvious whether you should use a less than or greater than sign, you should try the
limit
comparison test instead.
You must be able to show that your series is LESS than a KNOWN CONVERGING series to say
your series converges or GREATER than a KNOWN DIVERGING series to say your series diverges...
(You cannot tell what
happens if your series grows faster than a converging series or slower than a diverging series.)
8.
LIMIT COMPARISON TEST.
(GROWTH TEST).
If
n
a
is the series you don’t know about and you want to compare
it to a known series
n
b
,
Take
lim
n
n
n
a
b
r
°
to see which series GROWS faster.
(If the limit is infinity the top grows faster, if
the limit is zero the bottom grows faster, and if the limit is any other number they grow at the same rate.)
You must be able to
show that your series is growing SLOWER or at the same rate as the KNOWN CONVERGING series to say your series
converges; or show your series grows FASTER or at the same rate as the KNOWN DIVERGING series to say your series
diverges...
(You cannot tell what happens if your series grows faster than a converging series or slower than a diverging series.)
9.
FACTORIAL SERIES:
1
!
n
r
CONVERGES.
10.
ALTERNATING SERIES TEST (A.S.T.):
Alternating series converge if the following two conditions are true:
1
n
1.
lim
0
2.
<
a
(terms are decreasing).
n
n
n
a
a
+
=
OPTIONAL:
To save time, instead of using the A.S.T, we could check to see if a series ABSOLUTELY converges (i.e., the
absolute value of the series converges).
If the absolute value converges, then so will the alternating series, since when we put in
some subtraction signs in it will make our sum smaller
than the converging one!
If the series does NOT absolutely converge, we
would still need to use the A.S.T.
(Absolute convergence will not be on the exam specifically… it is only used as a time
saver if you want to use it!)
11.
RATIO TEST.
Take
1
lim
n
n
n
a
a
+
°
.
If the limit is LESS than one, the series CONVERGES.
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 Spring '08
 HanniNichols
 Mathematical Series, series A

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