Sequences & Series

Sequences & Series

1. Series

   • A series in math is the sum of the infinite terms in a sequence.
     • For example:
       • 1 + 3 + 5 + 7 + ...
     • Partial Sum
       • A partial sum when we sum up just part of a sequence, or in other words, just some of the terms.
         • For example (keeping in mind the last example given):
           • Partial Sum of first 3 terms: 1 + 3 + 5

2. Sequences

   • A sequence in math is a list of things (usually numbers) that are in order.
     • For example:
       • 3, 5, 7, 9,...
       • April, May, June, July,...

3. Arithmetic Sequences

   • In an arithmetic sequence the difference between one term and the next is a constant. In other words, we just add the same value each time infinitely.
     • For example:
       • 1, 4, 7, 10, 13, 16, ...
         • In this case, sequence has a difference of 3 between each number. The pattern is continued by adding 3 to the last number each time.
           • • Fuente: mathsisfun.com
     • In general we could write an arithmetic sequence like this:
       • {a, a+d, a+2d, a+3d, ... }

4. Geometric Sequences

   • In a geometric sequence each term is found by multiplying the previous term by a constant.
     • For example:
       • 1, 2, 4, 8, 16, ...
         • This sequence has a factor of 2 between each number. Each term (except the first term) is found by multiplying the previous term by 2.
           • • Fuente: mathsisfun.com
     • In General we write a geometric sequence like this:
       • {a, ar, ar2, ar3, ... }

5. Geometric Series

   • A geometric series would have the same structure as a geometric sequence, but in this case we sum all the terms.
     • For example:
       • {a + ar + ar2 + ar3 + ... }

6. Mathematical Induction

   • Mathematical Induction is a special way of proving things. It has only 2 steps:
     • Step 1:
       • Show it is true for the first one
     • Step 2:
       • Show that if any one is true then the next one is true
   • For example:
     • In the world of numbers we say:
       • Step 1: Show it is true for first case, usually n=1
       • Step 2. Show that if n=k is true then n=k+1 is also true

7. The Binomial Theorem

   • A binomial is a polynomial with two terms, so the binomial theorem is what happens when we multiply a binomial by itself many times, which will inevitably create a pattern.

8. Made by:

   • Marah Escobar Mathews