Taylor ((birthday)) reflection

so my blog is a bit late but lets see yesterday was my birthday and the saints actually won the super bowl
wow.

since everyone feared the limit lesson of chapter i guess thats what ill reflect on

there are two types of limit equations
the ones that use rules and the ones that use a calculator

the ones that use rules have simple hints to memorize for solving
the only ones that use rules are the polynomial equations problems

memorize this

((the rules))
t- top lead co
b- bottom lead co

t=b then coefficients
t>b then infinity
t
if you get a problem with a limit that is a polynomial equation
use the rules.
each and every time

the other type of problem is the one that calls for the use of a calculator
every single problem with limits that is not a polynommial equation calls for the use of a calculator

all you have to do is plug in for n three different times with
100
1000
10000
then plug into calculator
record what each outcome is and decipher what the numbers are headed toward which will then be your answer





what i dont understand is the sigma notation lesson
can anyone give me simplified hints or steps for working the sigma exuations

Alicia's Reflection # 25

Alrighty so GOOO SAINTSS.... Im not really a big fan but im happy for them for making history!!! Okay well last week we learned Infinite sequences and series and Sums of Infinite Series.

13-4 Infinite Sequences and Series

*lim
n-infinity: if the degree of the top= the degree of the bottom, then the answer is the coefficients.

Example:

lim n^2+1/2n^2-3n = 1/2
n-infinity

*lim
n-infinity: if the degree of the top is > the degree of the bottom, then your answer is infinity

Example:

lim 7n^3/4n^2-5 = infinity
n-infinity

*lim
n-infinity: if the degree of the top is < the degree of the bottom, then your answer is 0

Example:

lim 5n^2/3n^3+7 = 0
n-infinity

***If no rules apply, then you have to use your calculator to find what the sequence is approaching.

13-5 Sums of Infinite Series

*they can only be found with a geometric serires where /r/<1

Formula: S= t1/1-r

Example: 9-6+4

r= -6/9= -2/3 geometric

/-2/3/<1

S= 9/(1-(-2/3))= 27/5

Example: Write .45 repeating as a fraction

45/100-1= 45/99= 5/11


What i could use some help with is recursive definitions and sigma notation!! Thanks :)

Chapter 13 Sequences and Recursive Definitions

Sequences

A sequence is simply a list of numbers.

There are two main types of sequences:
Arithmetic - where you add or subtract
Geometric - where you multiply
(*Note: division is considered Geometric. For example: If a sequence divides by three, it is considered to be multiplied by one-third.

Formulas to find a term:

Arithmetic

tn-t'+(n-1)d

n=term #

t'=first term

d=what you add

tn=term#_in sequence

Geometric

tn=t'∙r^(n-1)

r= what you multiply by

Recursive Definitions

A recursive Definition is a formula for a sequence that involves a previous term. [a(n-1)]

an= (an-1/3)

Reflection #25

YEAH CUHHH! The Saints just won that CUH!

So um I like the Colts, that's my favorite team and I'm pumped for the Saints for finally winning, I don't know why but I don't even care that the Colts lost. But here's something I remember learning. Sequences and Series:

formulas:

1. Arithmetic- tn=t1+(n-1)d

n=term # t1=first term d=what you add

2. Geometric- tn=t1*r^(n-1)

r= what you multiply by n=term # t1=first term d=what you add


Ex.
6, 12, 24,.....

This is a Geometric sequence so you'd use the formula tn=t1*r^(n-1)

You would have 6*2^(n-1)

If you wanted to know the 4th term you would plug the 4 into the n and this is what you would get.

6*2^(4-1)
6*2^(3)
6*8
48

Now one thing I don't remember as of right now isssss, everything we did last week, I don't really remember anything from last week right now.

Stephanie's Reflection

tn-t1+(n-1)d (arithmetic)
sn=(n(t1+tn))/2

• n is term number
• t1 is first term
• d is what you add
• tn is term number in the sequence

tn=t'∙r^(n-1) (geometric)
sn=(t1(1-r^n))/1-r

• r is what you multiply by

lim/n infinity

• if the degree of the top is equal to the degree of the bottom then the answer is the coefficient
• if the degree of the top is greater than the degree of the bottom the answer is infinity
• if the degree of the top is less than the degree of the bottom the answer is 0
• if the rules don't apply, use your calculator

for geometric sequences if the absolute value of r is less than 1 then it goes to 0

the sum of infinite series can only be found when a geometric sequence where the absolute value of r is less than 1

s=t1/1-r

• if the absolute value of r is not less than 1, the series diverges (doesn't approach a number)
• if the absolute value of r is less than 1, the series approaches a number

Amy's Reflection #25

13 -4 Infinite Sequences & Series

Rules:

lim
n (infinity)

(used to plug in large numbers)

1. if the degree of the top = the degree of the bottom then the answer is the coefficients

2. if the degree of the top is > the degree of the bottom = infinity

3. if the degree of the top is < the degree of the bottom is 0

* if the rules don't apply you will have to use your calculator to find what the sequence is approaching

* for geometric sequences if |r| < 1 then it goes to 0

Examples:

1. Rule #1

3n^3 + 5n^2 + 6n^4/2n^3 + 5n^4 = 6/5

2. Rule #2

lim cos (1/n)
n (infinity)

cos(1/100) = .99995
cos(1/1000) = 1
cos(1/10000) = 1

= 1

3. Rule #3

5n^2 + √n/3n^3 + 7 = 0

4. no rule

lim (-10)^n
n (infinity)

(-10)^100 = -10000
(-10)^1000 = -100000

= -(infinity)

13 - 5 Sum of Infinite Series

*can only be found when a geometric sequence where |r|<1

Formula: S = t1/1-r

*if |r| is not less than 1 we say the series diveges - doesn't approach a #

* if |r|<1 then we say the series approaches a #

Examples:

1. Find the sum of the infinite series: 9 - 6 + 4

(geometric)

r = -6/9 = -2

|-2/3| < 1

S = 9/(1 - (-2/3)) = 27/5

2. For what values of x does the series converges? : 1 + (x-2) + (x+2)^2 + (x-2)^3

|x-2/1| < 1

-1 < x < 1

1 < x < 3

ok i need help with the problems that involves sigma, so if anyone can help me with that that would be great...

Stephen's Reflection

Ok so this week we are on chapter 13. This chapter is kinda sorta easy for right now because we basically use the same formulas which are arithmetic and geometric. Im going to explain the formulas for series.

Arithmetic series: Sn=(n(t1+tn))/2

Geometric series: Sn=(t1(1-r^n))/1-r

Ex: Find the sum of the first 25 terms of the series...11+14+17+20+...

first you find tn which is t25 and use the arithmetic formula which is t1+(n-1)d so it will be 11+(24)(3)=83.

Then you plug in the arithmetic series formula: Sn=25(11+83)/2=1175 and that is your answer.

The only thing i have problems with is what do i do with n when they dont tell me how many terms there are?