Series/Se qannces
Series
Power Series
Sequences
=
I.
>Z& +
no
ln(t
—
I.
y ~ < 1
~)9~t
lien
Nth term
test
=0 Them
it
(1.4)
(-4.4)
itt
(4.4)
leant cosnditionietlly
(4-1)
~n+1
ii.
lJ,~es5~O1ieenit diverges
~ —,V Q~ < 1
t~’) =
(-~.4)
Lice ACT
(a)
=
A in I iesiie ~Csn verse,, cc
“5±oosrDNsssDiverejencs
ALT. Series
(h)
converges, cheek Abs. Convergence en sing
lilt
ANY pd 0 r tent
2.
Il—a)
lilt’s altenyn Poo. or Neg., net any Pon. Series ‘flats.
(a)
3,
WillonlybeAbn.Cnnveege,et orOlergerit
Not Alternating belt also not always Pon. or Peg.,
teet for Abe. Cos,vergenoe or diveeenee. Conditional
eonverge,ite
~ Ei~=
snntamioptinn
(4--c)
(-4-4)
lnssriniomnienerien~diverges
(‘4-4)
~ E
(_1)n
(1-4)
AlL~r~~s’t~ innrnsenis eerlee a
-_
Conditionally Converge
Identities/Substitution
Trig/l-Iypvrbolic Trig
Exp/log
— la2a2 a a =
sinO c_no = U
~a2 + 1a2a2 a a af tan6
± lee 0 a ~co
Derivatives
Trig
Inverse Trig
f (ama)
at
coca
=
(amn~ a)
Hyperbolic Trig
_________________
a
~—
~;:;~‘
Exponent mi/Log
j~— (a9
(sinh a) a coals a
(coals a)
=
cmi: a
=
ascii2 a
a
cc5 Inca
sin a
~(cos’a)
(tan a)
=
Sec2 a
~ (tani a)
=
~ (tanis 0)
f. (ccc a)
a
coca tana
~ (sec” 1 a)
=
[ (sech a)
= —
cccl: a tanis a
f~- (cscha)
= —
cschacotha
tan2 0
(cotha)
a —
csch2 a
cot2 0
~ (cos a)
~‘
~.
f~-
= —
(ccc a) a
(cota)
—
= —
ccc a cot a
f
csc2 a
f~. (coc’ a)
~-,a>0
—
02
a a
a
~-
a
Inca 1
ccc 0
sinh=
cosh= ~ (c~ +c’~)
sin2 0 + cos~ 0
f__
(set”” a) a
‘1 la’~a2
~1,Ona)
1
Trig
Limits
÷ 1 = sec2 0
÷ 1 = csc2
mini
c—so
— =
=
-
inn
a
1
1—coca
a
c—so
=1
Integrals
Trig
Inverse Trig
f cosadu
=sina+c
fain ada=—coaa-l-c
f tan ada
a
In Icecal + c
= in I sec a ÷ tan UI +
fcocssda=lnicscu—cotal+c
fcotacia = ml sinai +c
f
~_i_da
f
I
—72—~-da
= ~.
nsf
da
=
Hyperbolic Trig
fsinhadaacosha-l-c
1. sin”’ (*) +
f cosh ada = sinh a ÷
f tanh ada = In(cosls a) ÷
tan~ (*) +
= ~-
aec1 (a) +
Nthterenteet,if
end
Miscelinneone
f2rze~=
*~~lflLf
f .-y.--~da — ~— In
÷
f 5/a2 ÷ U2da = tv’”2 + a2 ÷
I v’”2 — o2da = ~ v’a2 — a2 —
f5/e2
—
5/c2
less
as,
f
—~gda
fcc’~da =
=
a
md. Forms
Inlal +c
tmn ca
f’~-
÷ is)
+
+c
Ina +
~-
In IU + ‘/a~ _c21 +c
u2da a ~j-Va2 —a2 -i-sin’”
o
10a1
~ro
e=o
0-co
co=
00-00
D~
00’ 00=00
02+02=02
~°°
&=0
= 02
-
\\
¶=co
~-oc =~~°
+ a2) + a
S
(a) -i-c
eOrd’mnnryConspnsrinon Teet(eeneftsifortrig):ifO<nn<lsetVn>N
1.
ii~be~ennveegee, endeen E”n
2.
il~a,t diverges, no does
~snconvergeealimeetOtfnningAST
liens0 =OitennvcrgenbntetiilneedtotentiorAbn. Convergence oruivergense.
—
1da E fa’
f aechatanhada a — seclsa+ of ac°da a (a — Oc’5 +c
f cccl: a coth ada = — cccli a + c fin ada = a In(a) — a +
f arch2 da=tanlsa+c
f ~L_da a in I Inal + c
f cscls2 da = — coth a ÷ a
Geometric Series: so ≠O:Enr’’’ =s+ne-+er2-’- ~.testya__
•
f
fsoclsada=tais’lsisshal+c fc55dac’ +c
f sec ada
I soc a tan UdU = Sec a + c
f CCC a cot ada = -. coo a +
f sec2 ada = tan a +
f 0002 ada = — cot a + C
f cot2 ada = —Z — cot a +
Expnnentini/Log
0
a
3. •5l4k~ the eqneese thee,.
or ONE, then it diverges
Power Series: Alwnty onse the Aheolnie Ratio ‘Pent.
•lategr,tlTestsiif(x)ie;pne.,rnntinnnneasednn,n—inereseing,onjet,sn)then,
\
E
•
an
esverges a
“~w
What a saint nnaken the in go to tO
ff(a)d’e converges
P~erieeTeit, 0~1~a’
an> 1eonvergee,ss~1divergen
—
n11Plo
—
a I
‘
Is) ~
(—ti)
—Nowplngi,s—iandstothenriginntlfsntnnolve lorthecoeivergesteset
•
LimnitComparieonTest;Anenmesn>0,be,>0and(j~n—~.L
—ii0<L<oo,tlien~annndbnennsverge ordiverge together
ifL=Osnd~bnennveegee(p-eerientent)then,~ 5n converges
—
5.
taO
2.
(-RR)
3.
Whole Real Line
iL woo, do the integrnl tent
_iorEnetc55ins_aaEnei(e_n)”
•
RatioTentsndARTs
-_
p < 1 converges
.p>Sor
lien
Eon
isa ‘cries ofpoa.ter,neendli,n!!±±~.ap
I ART p < 1 Absolntely converge,
.-2±i=noDivergee
AWFaI3ivergen
—
p5incontlneive~ARTaMoreteeting,,eeded
—
Steps to help eolve,
—
5-
Plngin n+l for nil
2.
moltiply by the rca iproeai of the original
3.
nimplify,nndnoivetogetp
mOnte to
factor
(2w)l factors to, (2et)(2n
—.
1)(2et
—
2)(2es
—
3)! lyon svan,ted to factor Lisle
2.
‘nentervne I (a-a, a+R)
3.
whole rend line