Infinite Sequences and Summation Notation (9.1) The relationship between the value of a term and its position in line

POD– SAT Prep Only one today.

Infinite sequence notation Terms in a sequence are known by their value and their position (the index). The first term is labeled a 1 and the index is 1. The second is a 2 and the index is 2. The n th term is a n and the index is n. The sequence with n terms is a 1, a 2, a 3, …, a n. An infinite sequence has an infinite number of terms.

Sequences as functions These terms can also be noted in point form, as a functional relationship: (1, a 1 ), (2, a 2 ), (3, a 3 ), …, (n, a n ) where the x-term represents the location (the index) and the y-term is the value. In this form, what is the domain for an infinite sequence?

n th term notation One way to indicate the n th term is {a n }. For example, {2 n } has an n th term of 2 n. 2 1, 2 2, 2 3, 2 4, etc. Function notation would look like f(n) = 2 n. The graph of the function would be unconnected dots: (1, 2), (2, 4), (3, 8), (4, 16), etc. (If we graphed these as points, would the graph of a sequence be continuous or discontinuous. Why? What are the domain and range of this set?)

n th term notation Try it. Find the 4 th and 10 th terms of {2 + (.1 n )} Notice how the index number of the term is tied to the value of the term.

n th term notation Try it. Find the 4 th and 10 th terms of {2 + (.1 n )} n = 4 a 4 = = n = 10 a 10 = = Does this sequence seem to have a limit? What would its limit be? How could you think of it?

n th term notation Try it. Find the 4 th and 10 th terms of {2 + (.1 n )} n = 4 a 4 = = n = 10 a 10 = = When a sequence has a limit, we say it “converges” on that limit. If there is no limit, we say the sequence “diverges.”

n th term notation Try it. Find the 4 th and 10 th terms of {4}

n th term notation Try it. Find the 4 th and 10 th terms of {4} n = 4 n = 10 This is a trick question. Every term is 4.

n th term notation Try it. Find the 4 th and 10 th terms of {n 2 /(n+1)}

n th term notation Try it. Find the 4 th and 10 th terms of {n 2 /(n+1)} n = 416/(4+1) = 16/5 n = 10100/(10+1) = 100/11 If this were graphed as a rational function, would it have an asymptote? If so, what would the asymptote be?

Listing terms List the first 8 terms of the sequence given by {n 2 /(n+1)} : We can use a sequence button. Your calculator needs information in a certain order: formula, variable, start, stop. LIST – OPS – seq(x 2 /(x+1), x, 1, 8) How would you look at all the terms? How could you get this into fraction form?

Graphing sequences Let’s graph it on calculators. {n 2 /(n+1)} Put calculators in SEQ mode, then hit Y=. What do you notice? Enter the formula for the sequence. We will use the u part. nMin = 1, since we start with the first term. On the next line, enter the formula.

Graphing sequences Let’s graph it on calculators. {n 2 /(n+1)} Set the window to graph the first 8 terms. How does the graph correspond to what we think the asymptote would be?

Explicit and recursive formulas When we have the formula for the n th term, it is the explicit formula for the sequence (because it tells you explicitly what the term is). Some formulas tell you what a term is based on the previous term-- these are recursive formulas. Think of the Fibonacci Sequence: a 1 = 1 a 2 = 1 a n = a n-2 + a n-1 1, 1, 2, 3, 5, 8, 13, 21, etc.

Explicit and recursive formulas For example: This gives you the first term, and sets you up to find the following terms in succession. To find the next term, multiply the current term by 2. We could also write it What would an explicit formula be?

Explicit and recursive formulas For example: This gives you the first term, and sets you up to find the following terms in succession. 3, 6, 12, 24, … Or a n = 3(2) n-1

Partial Sums Sometimes you need to add the elements in a sequence. If we add only some of the terms in an infinite sequence, we call this a partial sum. Its notation: S 1 = a 1 S 2 = a 1 + a 2 S 3 = a 1 + a 2 + a 3 etc.

Summation notation Summation notation is very useful with partial sums. We use the Greek symbol sigma: means that we add the first n terms, or a 1 + a 2 + a 3 + … + a n. Unlike the S n notation we just saw, sigma notation not only tells us how many terms to add, but also directs us to the values of those terms.

Summation notation = a 1 + a 2 + a 3 + … + a n. Think of it this way: The bottom number is the first term, the top number is the final term, and the formula out to the side tells you what each term is.

Summation notation Let’s use summation notation to find some partial sums. How many terms are we adding in each one?

Summation notation Let’s use summation notation to find some partial sums.

Summation notation There are some special relationships with sums. Can you explain why these work?