Chapter 13: Sequences & Series

1. Chapter 13: Sequences & Series L13.4 Limits of an Infinite Sequence

2. Review of Rational Function Rules for Horizontal Asymptotes • Recall that is a rational function • They may have vertical, horizontal and slant asymptotes • In studying limits, we are interested in the horizontal asymptote because this is the value the function approaches when x gets very big • Rules for Horizontal asymptotes compare the degree of the numerator and the degree of the denominator: • If degree(N(x)) < degree(D(x)) HA is y = 0 • If degree(N(x)) = degree(D(x)) HA is y = ratio of leading coefficients • If degree(N(x)) > degree(D(x)) There is no HA Examples: a) b) c)

3. Limit of a Sequence • In every day language, a limit suggests a “barrier” • In Mathematics, a limit is a target that something approaches, but may not reach • Limits are an important part of Calculus • In this lesson, we consider the limits of infinite sequences, i.e., sequences that do not have a final term • The limit is the value that the sequence seems to approach as the index, n, goes to infinity

4. Heuristic approaches Algebraic approaches Limit of a Sequence How to determine the limit of a sequence: • Graph the sequence and visually inspect where the terms appear to be going • Observe the progression of terms to determine where the values appear to be going • Using the sequence’s formula, substitute a large value for n and compute the term’s value • If the formula is an exponential, rn, consider the value of r • If the formula is a rational expression, • algebraically manipulate the formula, or • invoke the rules for horizontal asymptotes Recall that sequences are discrete points that sit on corresponding function curves

5.   value of tn   1 2 3 4 5 6 7 . . . value of n The limit of as n goes to infinity is zero. Graph the Sequence As n gets larger and larger, is there a fixed number that the sequence values seem to be approaching?

6. Observe the Progression of Terms As n gets larger and larger, is there a fixed number that the sequence values seem to be approaching? The values are getting closer to 2.

7. Substitute a Large n Value 1, ½, ⅓, ¼, …. 1. .000001, … 2. 1, ¼, 1/16, 1/25, …. .000000000001, …

8. 2 tn 1 1 2 3 4 5 6 7 . . . value of n Substitute a Large n Value or Observe Graph

9. . . . . . . . . . . . . Formula has the form rn, |r| < 1 1. 2. If |r| < 1, the limit of the sequence is 0. Even if −1 < r < 0, the sequence will approach from both directions, homing in on zero.

10. Formula is a Rational Expression (1/3) Manipulate algebraically. Multiply top and bottom by 1/n3 The 6/n2 and 2/n3 terms go to zero as n→∞. So this expression approaches 0/1 as n→∞. Alternatively, Recall the Horizontal Asymptote (HA) Rules for Rational Functions. The degree in the denominator is greater than the degree in the numerator, so this expression has a HA of y = 0.[The discrete points for this sequence sit on the curve of the corresponding rational function. The HA is the graphical representation of the limit].

11. Formula is a Rational Expression (2/3) Manipulate algebraically. Multiply top and bottom by 1/n2 The 3/n in the denominator goes to zero as n→∞. So this expression approaches -1 as n→∞. Or, using the Horizontal Asymptote (HA) rules, the degrees in the top & bottom are the same so the HA that the curve approaches is the ratio of the leading coefficients. In this case, it is y = -1.

12. Formula is a Rational Expression (3/3) Manipulate algebraically. Multiply top and bottom by 1/n The 4/n and 3/√n go to zero as n→∞. So this expression approaches −2/5 as n→∞. Using the Horizontal Asymptote (HA) rules, the degrees are the same, so the ratio of the leading coefficients would prevail: −2/5.

13. 1 value of tn value of n 1 2 3 4 5 6 -1 NOT ALL SEQUENCES HAVE A LIMIT Consider this sequence : 1, - 1, 1, - 1, 1, . . . tn = (-1)n If the terms of a sequence do not “home in” on a single value, the sequence has no limit. [The limit ‘does not exist’ or ‘dne’]

14. 1 Value of tn 0 -1 value of n What is the Limit of the Sequence? The sequence has no limit. Note that if the function was missing the (-1)n, it would have a limit. What would it be?

15. Formula has the form rn, |r| > 1 1. 3, 9, 27, 81, . . . 59049, … -3, -9,-27, -81, . . . -59049, … 2. 3. -3, 9, -27, 81, . . . 59049, … If |r| > 1, the limit of the sequence does not exist (dne). The sequence increases or decreases without bound or toggles between these extremes as n → ∞. Note: When we say a limit = ±∞, the limit does not exist [∞ is not a real number].

16. Formula is a Rational Expression w/ no HA Manipulate algebraically. Multiply top and bottom by 1/n2 The 2 terms in the denominator go to zero as n→∞. So this expression approaches -∞ as n→∞. This sequence has no limit (∞ is not a real number). Since it is decreasing w/o bounds, it is said to be -infinity. Using the Horizontal Asymptote (HA) rules, the degree in the numerator is greater, so this expression has no HA (it increases or decreases without bounds).

17. Formula is a composed function Find limit of inside function: = 10 Therefore, = 1 When there is a composed function, find the limit of the inside function that has the variable, n. Then evaluate the outside function at that limit value (if it exists).

18. Limit of a Sequence: Summary • What value does the sequence seem to approach as the index, n, goes to infinity? • Heuristically determined: • Graph the points • Observe the progression of the sequence • Generate a term farther out, like t1000 • Algebraically determined: • If the formula for tn is an exponential expression, rn, if |r|<1, the limit is 0. Otherwise it grows to ±∞ or dne. • If the formula for tn is a rational expression, manipulate the exponents or use your knowledge of rational functions to determine the horizontal asymptote. • Limits do not exist if the expression does not “home into” a specific value. • If the sequence increases or decreases without bounds, we say its limit is positive or negative ∞, but in reality, it does not exist (dne). • If the expression involves a composed function, evaluate the limit of the inside function with the variable, and then evaluate the outside function with that limit value (if it exists).