bonfire09 said:Homework Statement
Let Ʃ 1\(n^2-1) from n=2 to k. It says find a closed for it and prove it using sum notation.
Homework Equations
The Attempt at a Solution
I can easily prove it by induction but I don't know what a closed form means. I tried looking it up online but there really isn't much info and nothing is stated in my textbook about. All I know is the sequence when expanded looks like
1/3+1/8+1/15+1/24+...+1/n^2-1. Then not sure how to put it in closed form?
bonfire09 said:This is what I get[tex] \left( \sum_{n=0}^k\frac{1}{-2(n+1)}+\frac{1}{2(n-1)} \right)=\frac{1}{3}+\frac{1}{8}+...+\frac{1}{-2(k+1)}+(\frac{1}{2(k-1)})[/tex]. From here I am not sure how to get a closed sum.
bonfire09 said:Yes but all I'm left is 1+1\2?
A closed form for a sequence is a mathematical expression that can be used to find any term in a sequence without having to calculate all the previous terms. It is also known as an explicit formula or an explicit definition.
A recursive formula defines each term in a sequence in terms of the previous terms, while a closed form expresses each term in a sequence as a function of the term number. This means that a closed form can be used to find any term in a sequence without having to know the previous terms.
Using a closed form for a sequence can save time and effort in calculating each term individually. It also allows for easier analysis and understanding of the sequence, as well as the ability to make predictions about future terms.
Not all sequences have a closed form. Some sequences are too complex or do not follow a specific pattern that can be expressed in a simple formula. In these cases, a recursive formula or another method may be used instead.
To find a closed form for a sequence, you can start by looking for patterns in the sequence and then try to express those patterns in a mathematical formula. You can also use techniques such as finite differences or generating functions to help find a closed form for a sequence.