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Infinte series formula sheet, Cheat Sheet of Mathematics

Series formulas are arithmetic’s and geometric series, special power series, Taylor and McLaurin series.

Typology: Cheat Sheet

2021/2022

Uploaded on 02/07/2022

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Series Formulas
1. Arithmetic and Geometric Series
Definitions:
First term: a1
Nth term: an
Number of terms in the series: n
Sum of the first n terms: Sn
Difference between successive terms: d
Common ratio: q
Sum to infinity: S
Arithmetic Series Formulas:
(
)
1
1
n
a a n d
=+−
1 1
2
i i
i
a a
a
+
=
1
2
n
n
a a
S n
+
=
(
)
1
2 1
2
n
a n d
S n
+
=
Geometric Series Formulas:
1
1
n
n
a a q
=
1 1
i i i
a a a
+
=
1
1
n
n
Sq
=
(
)
1
1
1
n
n
a q
Sq
=
1
1
1 1
fo
a
Sqr q
< <
=
2. Special Power Series
Powers of Natural Numbers
( )
1
1
1
2
n
k
k n n
=
= +
( )( )
2
1
1
1 2 1
6
n
k
k n n n
=
= + +
( )
2
3 2
1
1
1
4
n
k
k n n
=
= +
Special Power Series
( )
2 3
1
1 . . . 1 1
:
1
x x x x
x
for
= + + + + < <
( )
2 3
1
1 . . . 1 1
:
1
x x x x
x
for
= + + < <
+
2 3
1 . . .
2! 3!
xx x
e x= + + + +
( ) ( )
2 3 4 5
ln 1 . . . 1 1
2 3 4 5
:
x x x x
xorx xf
+ = + + < <
3 5 7 9
sin . . .
3! 5! 7! 9!
x x x x
x x= + +
2 4 6 8
cos 1 . . .
2! 4! 6! 8!
x x x x
x= + +
3 5 7
2 17
tan . . .
3 15 315 2 2
:for
x x x
xx x
π π
= + + + + < <
3 5 7 9
sinh . . .
3! 5! 7! 9!
x x x x
x x= + + + +
2 4 6 8
cosh 1 . . .
2! 4! 6! 8!
x x x x
x= + + + +
3 5 7
2 17
tan . . .
3 15 315 2 2
:for
x x x
x x x
π π
= + + < <
pf3

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Series Formulas

1. Arithmetic and Geometric Series

Definitions:

First term: a 1 Nth term: an

Number of terms in the series: n

Sum of the first n terms: Sn Difference between successive terms: d

Common ratio: q Sum to infinity: S

Arithmetic Series Formulas:

an = a 1 (^) + (^) ( n − (^1) ) d

1 1 2

i i i

a a a −^ +

1 2

n n

a a S n

(^2 1) ( 1 )

2

n

a n d S n

Geometric Series Formulas:

1 1

n a n a q

− = ⋅

ai = ai (^) − 1 ⋅ ai + 1

1 1

n n

a q a S q

1 (^1 )

1

n

n

a q S q

1

1

fo 1 1

a S q

rq

2. Special Power Series

Powers of Natural Numbers

( ) 1

n

k

k n n

∑^ =^ +

( )( )

2

1

n

k

k n n n

∑^ =^ +^ +

( )

3 2 2

1

n

k

k n n

∑^ =^ +

Special Power Series

( )

x x x x x

= + + + + for − < < −

( )

x x x x x

= − + − + for − < <

2 3 1... 2! 3!

x x^ x e = + x + + +

( ) ( )

2 3 4 5 ln 1... 1 1 2 3 4 5

x x x x

  • x = x − + − + for − < x <

3 5 7 9 sin... 3! 5! 7! 9!

x x x x x = x − + − +

2 4 6 8 cos 1... 2! 4! 6! 8!

x x x x x = − + − +

3 5 7 2 17 tan... 3 15 315 2 2

for :

x x x x x x

 π^ π

3 5 7 9 sinh... 3! 5! 7! 9!

x x x x x = x + + + +

2 4 6 8 cosh 1... 2! 4! 6! 8!

x x x x x = + + + +

3 5 7 2 17 tan... 3 15 315 2 2

for :

x x x x x x

 π^ π

3. Taylor and Maclaurin Series

Definition:

( )

( ) ( )

( )

2 1 1 ( )( ) ( ) ( ) ( ) ( )... 2! 1!

n^ n

n

f a x a^ f^ a^ x^ a f x f a f a x a R n

− − ′′ (^) − − = + ′ − + + + + −

( ) ( )( )

( ) ( )( ) ( )

( )

1

n^ n

n

n n

n

f x a R Lagrange s form a x n

f x x a R Cauch s form a x n

This result holds if f(x) has continuous derivatives of order n at last. If lim (^) n 0 n

R

→∞

= , the infinite series obtained is called

Taylor series for f(x) about x = a. If a = 0 the series is often called a Maclaurin series.

Binomial series

( )

1 (^ )^2 2 (^ )(^ ) 3 3

1 2 2 3 3

n (^) n n n n

n n n n

n n n n n a x a na x a x a x

n n n a a x a x a x

− − −

− − −

Special cases:

( )

(^1 2 3 ) 1 x 1 x x x x ... 1 x 1

  • = − + − + − − < <

( )

(^2 2 3 ) 1 x 1 2 x 3 x 4 x 5 x ... 1 x 1

  • = − + − + − − < <

( )

(^3 2 3 ) 1 x 1 3 x 6 x 10 x 15 x. .. 1 x 1

  • = − + − + − − < <

( )

1 2 1 1 3^2 1 3 5^3 1 1 ... 2 2 4 2 4 6

x x x x 1 x 1

− ⋅^ ⋅^ ⋅

( )

1 2 1 1 2 1 3^3 1 1 ... 2 2 4 2 4 6

x x x x 1 x 1

Series for exponential and logarithmic functions

2 3 1 ... 2! 3!

x x^ x e = + x + + +

( ) ( )

2 3 ln ln 1 ln ... 2! 3!

x x^ a^ x^ a a = + x a + + +

( )

2 3 4 ln 1 ... 2 3 4

x x x

  • x = x − + − − < x

( )

2 3 1 1 1 1 1 ln 1 ... 2 3

x x x x x x x

x

 −^   −^   − 