Series Formulas

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Series Formulas

1. Arithmetic and Geometric Series 2. Special Power 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:

an = a1 + (n -1) d

ai

=

ai -1

+ ai+1 2

Sn

=

a1

+ an 2

n

Sn

=

2a1

+ (n -1) d

2

n

Geometric Series Formulas:

an = a1 qn-1

ai = ai-1 ai+1

Sn

=

an q - a1 q -1

( ) Sn

=

a1

qn -1 q -1

S = a1 1- q

for -1 < q < 1

Powers of Natural Numbers

n k = 1 n (n +1)

k =1

2

n k 2 = 1 n (n +1)(2n +1)

k =1

6

n k 3 = 1 n2 ( n +1)2

k =1

4

Special Power Series

1 = 1+ x + x2 + x3 + . . . ( for : -1 < x < 1)

1- x

1 = 1- x + x2 - x3 + . . . ( for : -1 < x < 1)

1+ x

ex =1+ x + x2 + x3 + . . . 2! 3!

ln(1+ x) = x - x2 + x3 - x4 + x5 . . . ( for : -1< x ................
................

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