Kelly P.
ICm
Kelly's Tutor page
Chapter 12
Sequences and series
Section 3
infinite sequences and series
* Infinite sequence: a sequence with an infinite number of terms ( a sequence that goes on and on, with no end)
*Infinite series: the needed sum of the terms of an infinite sequence
The key to finding the limit:
Remember:
(2/n): as n increases the term decreases (i.e. when n gets bigger the term will approach zero!)
? = infinity
Lim
5n+15/n
n-->?
To find the Limit:
DIVIDE EVERYTHING BY THE HIGHEST N EXPONENT ON THE BOTTOM
* In this case the highest n exponent on the bottom is just n
* Divide everything by n:
5n/n +15/n /n/n
*The n's will cancel to give us:
5+15/n
* If you remember what I told you before, you will know that as n gets bigger it will approach zero. So, knowing that, make 15/n a 0! We are left with just 5! So, the limit is 5.
* If you remember to always divide by the highest exponent on the bottom you can do any problem!
Let's do a harder one:
It will be in the form of a word problem.
Bobby was having a hard time with his homework and needed major help. He asked his parents to help him, but they couldn't. They said they would try to find him a tutor, but it would probably take a week or two. Bobby couldn't wait that long so he got online to try to find some help. He found a website that "tutored" him by him typing in the problem and sending it to ask for help. He typed in the problem and sent it off. His problem looked as follows: n2 + 4n+6/ n2+2. Help Bobby do his homework.
* What is the highest exponent?
Right...n2
* What is the next step?
Yup... Divide everything by the highest exponent. So, the problem will look like:
n2/n2 +4n/n2 +6/n2 / n2/n2+ 2/n2
* Show Bobby what the new equation will look like with everything canceled.
1+4/n+6/n2 / 1+ 2/n2
* What do we know about what terms do as n increases?
Correct.... They approach zero. So, everything will be zero that is over an n. We are left with 1/1 or just 1. The limit is 1.
* Bobby now understands his homework!
Here is another thing to remember:
S= a1/ 1-r
S=sum A1= first term r= ratio
* You use this equation to find the sum of the infinite geometric series. ( Only when the absolute value of r is less than 1)
* You owe someone money, but don't have enough that day to pay him back completely. So, you make a deal with him that you will give him $10.00 the first day. And from each day forward you pay him half of what you paid him the day before. How much will you eventually give him?
* The series will look like this:
$10.00+$5.00+$2.50+...
We know the first term and the ratio is 1/2.
S= 10/ 1-1/2
s= 20
* we now know that he you will eventually pay someone $20.00!
"Teachers open the door, but you must enter by yourself."
~Chinese proverb
http://www.quoteland.com/
Teachers will always be willing to teach, but you have to come to class to learn. (Good Judgment)
For more on infinite series and sequences
http://www.math.vanderbilt.edu/~pscrooke/math155b/Chapters/Chapter12.pdf