Sequences and Series Formulas and Notes

Arithmetic Sequences and Series

An arithmetic sequence is a sequence where the common difference (d) between consecutive terms is constant.
d=T_{2}-T_{1}=T_{3}-T_{2}=T_{n}-T_{n-1}
e.g 4, 8, 12, 14, 20

The general term of an arithmetic sequence is
T_{n} = a+(n-1)d
a: first term.
d: common difference.

Example
For the sequence: 5, 9, 13, 17
a = 5.
d = T_{2}-T_{1} = 9-5 = 4.
T_{n} = a+(n-1)d
T_{n} = 5+(n-1)4
T_{n} = 4n+1

The formula for the series of an arithmetic sequence is
S_{n}= \frac{n}{2} \left[ 2a+(n-1)d \right]
a: first term.
d: common difference.

Example
For the sequence: 5, 9, 13, 17
a = 5.
d = T_{2}-T_{1} = 9-5 = 4.
S_{n}= \frac{n}{2} \left[ 2a+(n-1)d \right]
S_{n}= \frac{n}{2} \left[ 2(5)+(n-1)(4) \right]
S_{n}= \frac{ n \left( 4n+6 \right) }{2}

Geometric Sequences and Series

A geometric sequence is a sequence where the common ratio (r) between consecutive terms is constant.
r= \frac{T_{2}}{T_{1}}=\frac{T_{3}}{T_{2}}=\frac{T_{n}}{T_{n-1}}

The general term of a geometric sequence is
T_{n} = ar^{n-1}
a: first term.
r: common ratio.

Example
For the sequence: 4, 12, 36, 108
a = 4.
r = \frac{T_{2}}{T_{1}} = \frac{ 12 }{ 4 } = 3.
T_{n} = ar^{n-1}
T_{n} = 4 \cdot 3^{n-1}

The series formula of a geometric sequence is
S_{n}= \frac{a(r^{n}-1)}{r-1} for r > 1 OR

S_{n}= \frac{a(1-r^{n})}{1-r} for r < 1
a: first term.
r: common ration.

Example
For the sequence: 4, 12, 36, 108
a = 4.
r = \frac{T_{2}}{T_{1}} = \frac{ 12 }{ 4 } = 3.
r is greater than 1, so we use S_{n}= \frac{a(r^{n}-1)}{r-1}
S_{n}= \frac{a(r^{n}-1)}{r-1}
S_{n}= \frac{4(3^{n}-1)}{3-1}
S_{n}= \frac{4(3^{n}-1)}{2}
S_{n}= 2 \cdot 3^{n}-2

Example
For the sequence: 81, 27, 9, 3
a = 81.
r = \frac{T_{2}}{T_{1}} = \frac{ 27 }{ 81 } = \frac{1}{3}.
r is less than 1, so we use S_{n}= \frac{a(1-r^{n})}{1-r}
S_{n}= \frac{a(1-r^{n})}{1-r}
S_{n}= \frac{81(1- \left( \frac{1}{3} \right)^{n})}{1- \frac{1}{3}}
S_{n}= \frac{81- 81 \cdot \left( \frac{1}{3} \right)^{n}}{\frac{2}{3}}
\displaystyle S_{n}= \frac{243}{2} - \frac{243 \cdot 3^{-n}}{2}

Quadratic Sequences

A quadratic sequence is a sequence where the the second difference is common. Below is an example.

General Term

The general term of a quadratic sequence is
T_{n} = an^{2}+bn+c
then 2a = second difference.
3a+b = T₂-T₁
a+b+c = first term.

Example
For the sequence: (The example above) 6, 12, 22, 36
Second Difference = 4.
General term: T_{n} = an^{2}+bn+c. We need to get a, b and c.
2a = Second Difference
2a = 4
\bullet a = 2
3a+b = T_{2}-T_{1}
3(2)+b = 12-6
\bullet b = 0
a+b+c = T_{1}
(2)+(0)+c = 6
\bullet c = 4
\displaystyle T_{n} = 2n^{2}+4

Taylor Series

\sum_{n=0}^{\infty } \frac{f^{(n)}(a)(x-a)^{n}}{n!} = f(a) + \frac{f^{'}(a)}{1!}(x-a) + \frac{f^{''}(a)}{2!}(x-a)^{2} + \frac{f^{'''}(a)}{3!}(x-a)^{3}+...

Example
\displaystyle f(x) = e^{5x}, at a = 1
\displaystyle f^{'}(x) = 5e^{5x}
\displaystyle f^{''}(x) = 5^{2}e^{5x}
\displaystyle f^{'''}(x) = 5^{3}e^{5x}
\displaystyle ...
Taylor series: f(a) + \frac{f^{'}(a)}{1!}(x-a) + \frac{f^{''}(a)}{2!}(x-a)^{2} + \frac{f^{'''}(a)}{3!}(x-a)^{3}+...

\displaystyle e^{5(a)} + \frac{ 5e^{5(a)} }{1!}(x-a) + \frac{ 5^{2}e^{5(a)} }{2!}(x-a)^{2} + \frac{ 5^{3}e^{5(a)} }{3!}(x-a)^{3}+...

At a = 1: \displaystyle e^{5(1)} + \frac{ 5e^{5(1)} }{1!}(x-1) + \frac{ 5^{2}e^{5(1)} }{2!}(x-1)^{2} + \frac{ 5^{3}e^{5(1)} }{3!}(x-1)^{3}+...

Maclaurin Series

\sum_{n=0}^{\infty } \frac{f^{(n)}(0)x^{n}}{n!} = f(0) + \frac{f^{'}(0)}{1!}(x) + \frac{f^{''}(0)}{2!}x^{2} + \frac{f^{'''}(0)}{3!}x^{3}+...

Example
\displaystyle f(x) = \frac{1}{1-x}
\displaystyle f^{'}(x) = \frac{1}{(1-x)^{2}}
\displaystyle f^{''}(x) = - \frac{2}{(x-1)^{3}}
\displaystyle f^{'''}(x) = \frac{6}{(x-1)^{4}}
\displaystyle ...
Maclaurin series: f(0) + \frac{f^{'}(0)}{1!}(x) + \frac{f^{''}(0)}{2!}x^{2} + \frac{f^{'''}(0)}{3!}x^{3}+...

\displaystyle = \frac{1}{1-0} + \frac{ \frac{1}{(1-0)^{2}} }{1!}(x) + \frac{ - \frac{2}{(0-1)^{3}} }{2!}(x)^{2} + \frac{ \frac{6}{(0-1)^{4}} }{3!}(x)^{3}+...

\displaystyle = 1 + x + x^{2} + x^{3} + ...