Infinite Limits & Infinite Series

1. Infinite Limits & Infinite Series Mike Madro

2. Infinite Limits • Is a list or set of numbers written in definite order. • T1/2, t2/3, t3/4, t4/5, ..., tn,… • In an infinite sequence each term has a successor • As a sequence approaches a number such as 1 , we can say that the limit of the sequence is 1

3. Infinite Series • Is the adding of an infinite amount of numbers • In which we allow Sn to represent the sum of the terms in the series • Ex. Sn= 0.5+0.25+0.125+0.0625 • As we add more terms to the series the sum gets closer to 1, therefore we can assume that the sum of the infinite series is 1 • If a series has a sum, it is called a convergent series. If not, it is called divergent.

4. Real World • Zeno’s second paradox involves a race between the Greek hero Achilles and a tortoise. In this problem Zeno argues that at Achilles start position a1 and the tortoise start position t1, when Achilles reaches point a2=t1 the tortoise will be ahead at position t2. Therefore, at position a3=t2 the tortoise will still be ahead at t3. This process should continue indefinitely.

5. Real World Infinite Series • In another of Zeno’s paradoxes: a man is not able to walk into a wall because he would always be going half the distance. This halving of distance would continue indefinitely.

6. How to learn infinite limits • Sub positive integers into the place of n • Determine what the limit approaches

7. How to learn infinite series • Sub positive integers into the place of n • Add up the series

8. Examples =2

9. Questions • State the limits of the following sequence, or state that the limit does not exist. • 1/3, 1/9, 1/27, 1/81, 1/243, …, (1/3) 2. 1, 2, 3, 4, 5, …, n, … n

10. Find the following limit or state the limit does not exist. • 3) • 4) Lim Lim

11. Determine whether the following series converges or diverges 5) Lim

12. Find the sum of the following series or state that the series is divergent. • 6) 1+ 1/3 + 1/9 + 1/27 + … • 7) 1 – 2 + 4 – 8 + … • 8) • 9)

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