infinite limits & infinite series

Infinite Limits & Infinite

Series

Mike Madro

Infinite LimitsIs 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

Infinite SeriesIs 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 1If a series has a sum, it is called a convergent series. If not, it is called divergent.

Real WorldZeno’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.

Real World Infinite SeriesIn 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.

How to learn infinite limits1) Sub positive integers into the place of n2) Determine what the limit approaches

How to learn infinite series1) Sub positive integers into the place of n2) Add up the series

Examples=2

Questions State the limits of the following sequence, or state that the limit does not exist.

1. 1/3, 1/9, 1/27, 1/81, 1/243, …, (1/3)2. 1, 2, 3, 4, 5, …, n, …

n

Find the following limit or state the limit does not exist.3)

4)

Lim

Lim

Determine whether the following series converges or diverges

5) Lim

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)

10)