Ok, I want to take some time to make clear some simple definitions that might be confusing sometimes under sequence and series to help anyone who might need such definition at the last minute for a test or anything. I’m just going to define some important terms but don’t hold me responsible for defining everything about sequence and series. I’ll just state the term and give the definition. This can come in handy especially if you are taking a calculus class and you have to provide definition for some terms under series and sequences.
1. Sequence is a list of numbers A1, A2, A3,…, An in a given order. example 1,2,3,4,5
2. Sequence {an} converges to a number L if for every positive number E there corresponds an integer N such that for all n, n>N implies that |An – L| < E. if {an} converges to L, then limit as n turns to infinity is L or simply An -> L
3. Sequence {An} diverges to positive infinity if for every number M, there is an integer N such that for all n>N, An>M. In this case, the limit of An as n turns to infinity is simply infinity or An -> infinity.
4. Sequence {An} diverges to negative infinity if for every number M, there is an integer N such that for all n < N, An < M. In this case, the limit of An as n turns to infinity is negative infinity or An -> negative infinity.
5. Sequence {An} is bounded above if there exists a number M such that An<=M for all n.
6. Sequence {An} is bounded below if there exists a number M such that An >= M for all n.
7. Sequence {An} is bounded if there exists a number M such that An<=M for all n and there exists a number K such that An>=K for all n.
8. Sequence {An} is unbounded if there isn’t either a number M such that An<=M or a number K such that An>= K for all n.
9. The greatest lower bound of a sequence {An} is a number M which is a lower bound of {An} and there exists no number greater than M which is a lower bound of the sequence{An}.
10. The least upper bound of a sequence {An} is a number M which is an upper bound of {an} and there exists no number less than M which is an upper bound of the sequence {An}.
11. A sequence {An} is non-decreasing if An<=An+1 for all n.
12. A sequence {An} is non-increasing if An >= An+1 for all n.
13. Infinite Series is the sum o f an infinite sequence of numbers, A1+A2+A3,…,An+….
14. Partial sum of an infinite series is the sum of it’s first n terms for some n.
15. Geometric series are series of the form a+ar+ar^2+…+ar^n-1 = 
in which a and r are fixed real numbers and a is not equal zero. It can also be represented as 
which is like shifting the index. The sum of this series is simply a/(1-r) where r is not equal 1.
16. Harmonic series is a series of the form 
. This type of series diverges.
17. Telescoping series is a type of series such that the partial sum of the series is simply the sum of first and last terms. Other terms in the series cancel out with each other leaving only the first and last terms to compute the sum.
