A-level Mathematics/CIE/Pure Mathematics 1/Series

The Binomial Theorem edit

Before we discuss the binomial theorem, we need to discuss combinations. In order to discuss combinations, we need to discuss factorials.

Factorials edit

The factorial of a number is the product of all numbers from 1 to that number. It is represented by the symbol   after the number.

e.g.  

The factorial can be formally defined as:

 

Combinations edit

 
10 3-item combinations can be chosen from a set of 5. Thus, 5C3 = 10.

Combinations are a way of calculating how many ways a set of items with a given size can be selected from a larger set of items. It is typically represented either by the column notation   or by the notation  .

Combinations can be calculated using factorials:  .

e.g.  

The Binomial Theorem edit

The binomial theorem is used when we need to raise a binomial, an expression consisting of two terms, to the power of a given  , e.g.  .

The binomial theorem states that  

e.g.

 

The binomial theorem is sometimes summarised as  

Arithmetic Progressions edit

An arithmetic sequence is a progression in which the numbers increment by a fixed quantity from one term to the next.

e.g.   is an arithmetic sequence (the fixed quantity is  )

The nth term edit

The nth term of an arithmetic sequence can be determined using   where   is the nth term,   is the first term, and   is the difference between two consecutive terms in the progression.

 
A visual proof for how the sum of an arithmetic sequence can be found

e.g. The sequence   has a difference of  . So the nth term of this sequence can be determined by  . Thus, if we wanted to find the 1000th term of the progression, we can use the nth term formula:  .

Sum of the first n terms edit

The sum of the first n terms of an arithmetic progression can be found using the formula:  

e.g. Find the sum of the first 50 terms of the sequence  

 

Geometric Progressions edit

A geometric progression is like an arithmetic progression except that instead of adding a constant from one term to the next, we multiply each term by a constant to get the next term.

e.g.   is a geometric sequence.

The nth term edit

The nth term for a geometric progression is given by   where   is the nth term,   is the first term, and   is the ratio between two consecutive terms.

Sum of the first n terms edit

 
Proof without words of the formula for the sum of a geometric series – if |r| < 1 and n → ∞, the r n term vanishes, leaving S = a/1 − r

The sum of the first n terms of a geometric series can be found using  .

e.g. The sum of the first 10 terms of the sequence   is  .

Convergence edit

A convergent geometric progression is one where the terms get smaller and smaller, meaning that as   approaches infinity, the  th term approaches zero. An important consequence of this is that the progression will have a defined sum to infinity.

We can tell if a sequence is convergent if the ratio   is less than   and more than  . If this condition is not satisfied, the sequence is divergent.

Sum to Infinity edit

The sum to infinity of a geometric progression is the value that the sum of the first   terms as   approaches infinity. If a progression is convergent, its sum to infinity will be finite.

The sum to infinity is given by   which is equivalent to   if  .

e.g. The sum to infinity of the sequence   is  

Trigonometry · Differentiation