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Summary Introduction to Series | Calculus II Notes CA$12.26
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Summary Introduction to Series | Calculus II Notes
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An introduction to series, including definitions, relation to integration, theorems and detailed examples

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  • July 14, 2023
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  • 2022/2023
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3.2 Intro to Series
Definition:
series
A a sum of infinitely many terms;(a, + az azt...)
+




notation: ne, An

eg. 0neikn( k 4
=
+

6 4x+ ...)
+ +




② 0.333...
...
-
=




Bon
=




③ Definite integral! (...)
n
·
- (...)
k1 =




Given a series, mean, we define the nth partial sum of the series to be Sn aj
=

A,
=



taztAgt...
ifthe
sequence [Sn3 converges and **Sn=5 then we say the series,an converges to 5 and we

writen =,An S =




if ESn3 diverges, no,an diverges

eg. 0An= (-1)", e,)-1) = ?
Sn 2,) 1) 1 1 1 1...1)
S 0,
-
his even
-
= = + - +
=




-

, n is odd

↳MoSn dne, ESn] diverges

Ian ?,(-1)" =


diverges

② Suppose the
&
nth partial sum ofa series,an is
Sn=5-an
a) whatis n=,An? divide each termby n



hi Sn (5
=
-


) (S-3in
-
=
5-43 is
=




so ,an 1/3
=




(convergent)
b) Whatis an?
Sn a, Act
= +

. . . An-itAn

Sn-(Sn-1) An =




(
an=
($-3ny)-15-an-is
1) 3(n -
4
+




2(n 1)
-
-




3(n 1) 4
+
-




③Yo (1+
=
r +
r2 r3+ +
...)

Fr 1
=
-r* for rel
1- r

1
Sn 2 =un+1-1
+



1
for re
-

=




1 -
r r -
d

, inr
E ,if Ir)< otherwise
1
0 if and
diverges
=




r 1
=




&



eifum
e
1
fn
+




1
Iris
8
-




if
1 - r




ifIr)< 1 and
r diverges otherwise
-




nEor= I
if (t) <
1
"geometric series"
1 -
r

geometric sum
-
eg. itn n=o(t)" r tz
=

=




-I 2
=




1 -

t
·

In?
1
n =



Eit otr-Stermwhere =


n 0)
=




t
2 1 1
-
- -
- = =




② n=Yer no(ter) =




geometric series with Ir)= be:
e
- te
-
=




-2 -
1




③_E52 n=0(-43) =




-


8/3 1- 83) 8/31 =




Is
=




divergea



Another case,where you find
a formula for Sn:"telescoping sums"
eg. E, n(n+1)

Note
mentistint
Sn x=i)k
=
-

x 1) +




-

- et
substitute k+1 for t

~l
22

-Eite-e
=




e=2 to
Sn = 1 (for everything cancels) -
I

n+1


( + 1) 1
in =
(1
-
=




&


n ,n(n 1)
=
+




eg.n ·fi)n3
0
=




+3
-




=ni(ts)" 3(55) +

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