So, since i hardly ever ask questions..and always in need of knowledge. I'll ask many questions to hopefully be answered.
1. When do you know if a test fails you and you need to do something else.
2. What tests are "inconclusive"?
3. What does inconclusive mean?
4. What came first: the chicken or the egg?
5. How do you do series with trig functions..like cos(pi/x)..?
6. How do you do the word problem things?
For the most part, i understood this chapter..these six questions always confused me though.. and i think on the test today i began to become more worried about that than what i needed to be worried about!
Monday, November 1, 2010
Post #10
Sequence:
take the limit to see if it diverges or converges
+if it has a limit it converges
+if you have a type of infinity it diverges
Finding terms: plug into equation
Partial Sums: add the term before it
Series:
Arithmetic sum: n(t+tn)/2
Geometric sum: t1/1-r
nth term test:
take the limit, may have to use L"H rule
+if you get zero must use different test
+if you get a number it diverges
Integral Test:
if you can integrate it easily
+if x is greater than or equal to one it converges
+anything else diverges
p-series test:
1/n^p
+if p is greater than 1 it converges
+if it is less than or equal to 1 it diverges
Geometric:
+if absolute value of r is less than 1 it converges
+greater than or equal to one it diverges
direct comparison test:
compare it to something easier
you then might get a geometric or p-series
use the test that works
+if you get it converges, stop
+if it diverges you must know ?
Limit Comparison Test:
compare to something easier
use another test
+divide original by compared
+take limit, if +ve number it converges
+if infinity, it diverges
alternating series:
will have like (-1)^n+1
+take the alternating part out for new
+take limit, must get zero
*if you get a number it diverges
+now add +1 to each n, less than or equal to, new one
+if true converges
+if not true diverges
Ratio Test:
usually used with ! or variable exponents
+add +1 to each n divided by the original
+things should cancel
* remember 2^n+1 can be written has 2^n 2^1 and (n+1)! is like n!(n+1)
+take limit
+less than 1 it converges, greater than 1 or infinity it diverges
+if you get one it is inconclusive
root test:
used when something is raised to the n
+take the nth root of the absolute value of the original
+take limit
+if you get less than 1 it converges, if you et greater than one or infinity it diverges
+if you get one it is inconclusive
Absolute Convergence: if absolute value of an converges then an converges
Conditional Convergence: if an converges but absolute vale does not converge
take the limit to see if it diverges or converges
+if it has a limit it converges
+if you have a type of infinity it diverges
Finding terms: plug into equation
Partial Sums: add the term before it
Series:
Arithmetic sum: n(t+tn)/2
Geometric sum: t1/1-r
nth term test:
take the limit, may have to use L"H rule
+if you get zero must use different test
+if you get a number it diverges
Integral Test:
if you can integrate it easily
+if x is greater than or equal to one it converges
+anything else diverges
p-series test:
1/n^p
+if p is greater than 1 it converges
+if it is less than or equal to 1 it diverges
Geometric:
+if absolute value of r is less than 1 it converges
+greater than or equal to one it diverges
direct comparison test:
compare it to something easier
you then might get a geometric or p-series
use the test that works
+if you get it converges, stop
+if it diverges you must know ?
Limit Comparison Test:
compare to something easier
use another test
+divide original by compared
+take limit, if +ve number it converges
+if infinity, it diverges
alternating series:
will have like (-1)^n+1
+take the alternating part out for new
+take limit, must get zero
*if you get a number it diverges
+now add +1 to each n, less than or equal to, new one
+if true converges
+if not true diverges
Ratio Test:
usually used with ! or variable exponents
+add +1 to each n divided by the original
+things should cancel
* remember 2^n+1 can be written has 2^n 2^1 and (n+1)! is like n!(n+1)
+take limit
+less than 1 it converges, greater than 1 or infinity it diverges
+if you get one it is inconclusive
root test:
used when something is raised to the n
+take the nth root of the absolute value of the original
+take limit
+if you get less than 1 it converges, if you et greater than one or infinity it diverges
+if you get one it is inconclusive
Absolute Convergence: if absolute value of an converges then an converges
Conditional Convergence: if an converges but absolute vale does not converge
Maleries post
Okay. So I, for one, would like to know WHEN DOES THIS CHAPTER END??? I am BEYOND tired of sequences and series....
AHH.. okay. Math right? well perhaps this will help us in future endeavors.
Okay, so first, to see if a SERIES converges or diverges, you follow that chart.
First things first, nth term. you just take the limit as x goes to infinity. If you get anything other than zero, the series diverges right of the bat. If you get zero, the nth term test is inconclusive.
Next, you can determine whether or not it's a pseries, geometric, or something you can integrate.
PSERIES is when its n raised to some exponent
GEOMETRIC is when it's some fraction raised to the n
Integral is when its something easy..so say ln integration would be easy to do (i.e. 1/n)...or if you feel like messing with by parts, go for it.
Pseries-if p(exponent) is greater than one it converges, if its less than or equal to it diverges
Geometric-if abs(r) (your thing being raised) is less than one, it converges. Not it diverges.
Integral-if you get after integrating infinity anywhere, it diverges.# it converges
If the above do not apply you have a couple of options.
There's the ROOT TEST (where you just force a root...I'm a little shady on this one). RATIO TEST (where you add one to every n and put that over your original and take the limit) ALTERNATING SERIES TEST (which is a little tricky..check your book.) DIRECT COMPARISON TEST (compare it to something bigger and try to use pseries, geo, integral, etc.) and LIMIT COMPARISON TEST where you just compare it to something and put your original over what you're comparing it to and take the limit.)
AHH.. okay. Math right? well perhaps this will help us in future endeavors.
Okay, so first, to see if a SERIES converges or diverges, you follow that chart.
First things first, nth term. you just take the limit as x goes to infinity. If you get anything other than zero, the series diverges right of the bat. If you get zero, the nth term test is inconclusive.
Next, you can determine whether or not it's a pseries, geometric, or something you can integrate.
PSERIES is when its n raised to some exponent
GEOMETRIC is when it's some fraction raised to the n
Integral is when its something easy..so say ln integration would be easy to do (i.e. 1/n)...or if you feel like messing with by parts, go for it.
Pseries-if p(exponent) is greater than one it converges, if its less than or equal to it diverges
Geometric-if abs(r) (your thing being raised) is less than one, it converges. Not it diverges.
Integral-if you get after integrating infinity anywhere, it diverges.# it converges
If the above do not apply you have a couple of options.
There's the ROOT TEST (where you just force a root...I'm a little shady on this one). RATIO TEST (where you add one to every n and put that over your original and take the limit) ALTERNATING SERIES TEST (which is a little tricky..check your book.) DIRECT COMPARISON TEST (compare it to something bigger and try to use pseries, geo, integral, etc.) and LIMIT COMPARISON TEST where you just compare it to something and put your original over what you're comparing it to and take the limit.)
Sunday, October 31, 2010
post 10.
soooo, we got a big test tomorrow in calc. so let's go over what is gonna be on that!
p-series:
used whenever you have n^#.
if # is 1 it converges.
geometric series:
used whenever you have #^n
if abs.value of # <> 1, continue on to testing..
you would divide original by what you compared it to. then take the limit as n goes to infinty of that. & if you get # > 1, it diverges. if not, it is inconclusive.
nth term test:
you use this when you don't exactly know what else to do & you are just testing.
take the lim as x goes to infinity.
if you get anything besides 0, it diverges. if you get 0, it's inconclusive.
integral test:
this is used whenever you have absolutely nothing else to do. you take the integral from n to infinity
if you get infinity, it diverges. if you get a number, it converges.
alternating series test:
used whenever you have -1 or -2 raised to n.
you take out that portion of the problem, then continue on. but if you have -2, you just take out the negative.
after taking it out, you take the limit as n goes to infinity.
if you get anything besides 0, it diverges. if you get 0, continue on & add one to every n.
then compare to the original after you took out the -# ^ n by doing (n+1 equation) ^ if above is true, then converges. if false, then diverges.
sum of geometric series:
first term/1-r
sum of nongeometric series:
n(t1 + tn)/2
ratio test:
use if it tells you to.. pretty much.
add one to each n. then divide <-- that by the original. then take lim as n goes to inifinty.
if you get # <> 1 or infinity, it diverges.
(also you use this if you have a factorial!)
NOW for what I don't understand...
-direct comparison test
-limit comparison test
-absolute convergence
-conditional convergence
p-series:
used whenever you have n^#.
if # is 1 it converges.
geometric series:
used whenever you have #^n
if abs.value of # <> 1, continue on to testing..
you would divide original by what you compared it to. then take the limit as n goes to infinty of that. & if you get # > 1, it diverges. if not, it is inconclusive.
nth term test:
you use this when you don't exactly know what else to do & you are just testing.
take the lim as x goes to infinity.
if you get anything besides 0, it diverges. if you get 0, it's inconclusive.
integral test:
this is used whenever you have absolutely nothing else to do. you take the integral from n to infinity
if you get infinity, it diverges. if you get a number, it converges.
alternating series test:
used whenever you have -1 or -2 raised to n.
you take out that portion of the problem, then continue on. but if you have -2, you just take out the negative.
after taking it out, you take the limit as n goes to infinity.
if you get anything besides 0, it diverges. if you get 0, continue on & add one to every n.
then compare to the original after you took out the -# ^ n by doing (n+1 equation) ^ if above is true, then converges. if false, then diverges.
sum of geometric series:
first term/1-r
sum of nongeometric series:
n(t1 + tn)/2
ratio test:
use if it tells you to.. pretty much.
add one to each n. then divide <-- that by the original. then take lim as n goes to inifinty.
if you get # <> 1 or infinity, it diverges.
(also you use this if you have a factorial!)
NOW for what I don't understand...
-direct comparison test
-limit comparison test
-absolute convergence
-conditional convergence
Monday, October 25, 2010
Post #9
Well does it converge or diverge?
You should probably try the nth term test before trying another method.
P-Series: n^p
if p is greater than 1-->converge
if less than or eqaul to 1-->diverge
Geometric: (1/2)^n
if the absoultue value of r(which is the n) is less than 1-->converge
if the absoultue value of r greater than 1-->diverge
Intergral Test: Probably use when everything fails.
First, take the limit from 0 to infinity.
Second, integrate like normal.
Third, plug in 0 and infinity.
Finally, take the limit.
If you still get any form of infinity-->diverge
if you get a number-->converge.
Comparing: You can compare the problem to an eaiser one.
1. direct comparison
2. limit comparison
My Question: I need someone to explain the difference between direct and limit (the steps and everything). I'm afraid that I'm mixing them up or using a combination of both. Please help.
You should probably try the nth term test before trying another method.
P-Series: n^p
if p is greater than 1-->converge
if less than or eqaul to 1-->diverge
Geometric: (1/2)^n
if the absoultue value of r(which is the n) is less than 1-->converge
if the absoultue value of r greater than 1-->diverge
Intergral Test: Probably use when everything fails.
First, take the limit from 0 to infinity.
Second, integrate like normal.
Third, plug in 0 and infinity.
Finally, take the limit.
If you still get any form of infinity-->diverge
if you get a number-->converge.
Comparing: You can compare the problem to an eaiser one.
1. direct comparison
2. limit comparison
My Question: I need someone to explain the difference between direct and limit (the steps and everything). I'm afraid that I'm mixing them up or using a combination of both. Please help.
Sunday, October 24, 2010
A blog
So steph came to me and asked me how to break down the steps for finding whether or not a series converges or diverges. So...I believe they're the following...
1. nth term test-so basically you're going to just take the limit as n goes to infinity right off the bat...if it equals something other than zero, you know for sure that it diverges. However, If you get 0, you have to proceed to next step...
2. Okay. for step two you can choose between a, b, and c.
a. p series
b. geometric
c. integral test
**basically, you have to recognize that if it's 1/n^(exp), it's a pseries. If not you see if it's geometric (i.e. something raised to the n). And if it's something that looks easy to integrate, integrate it.
If step 2 fails, go onto step 3.
3. Unless the problem tells you otherwise, you can use either direct comparison OR limit comparison. For direct, you just compare it to something bigger (so if i have a fraction, the number on the bottom will be smaller...I know, confusing...). So once I find something bigger and similar, I just go through step 2. Same goes for limit comparison test, although, all you need is something similar, doesn't have to be either bigger or smaller than the original. Then all you do is take the limit as n goes to infinity.
simple enough??
I know it's confusing, you just have to think through all the steps.
QUESTIONS:
I have NO CLUE WHATSOEVER how to determine if a test is "inconclusive". So if you have that practice packet, problem 16...I got the integral test done, but I got infinity minus something. Does that mean it's inconclusive??? kbye.
1. nth term test-so basically you're going to just take the limit as n goes to infinity right off the bat...if it equals something other than zero, you know for sure that it diverges. However, If you get 0, you have to proceed to next step...
2. Okay. for step two you can choose between a, b, and c.
a. p series
b. geometric
c. integral test
**basically, you have to recognize that if it's 1/n^(exp), it's a pseries. If not you see if it's geometric (i.e. something raised to the n). And if it's something that looks easy to integrate, integrate it.
If step 2 fails, go onto step 3.
3. Unless the problem tells you otherwise, you can use either direct comparison OR limit comparison. For direct, you just compare it to something bigger (so if i have a fraction, the number on the bottom will be smaller...I know, confusing...). So once I find something bigger and similar, I just go through step 2. Same goes for limit comparison test, although, all you need is something similar, doesn't have to be either bigger or smaller than the original. Then all you do is take the limit as n goes to infinity.
simple enough??
I know it's confusing, you just have to think through all the steps.
QUESTIONS:
I have NO CLUE WHATSOEVER how to determine if a test is "inconclusive". So if you have that practice packet, problem 16...I got the integral test done, but I got infinity minus something. Does that mean it's inconclusive??? kbye.
post 9
so, still confused with chapter 9 stuff..
i'll just go over what i know.
so if you are given something, (idk if it matters if it is a sequence or a series).. you automaticaly take the nth term test.
so take lim as n goes to infinty. if you get 0, you have to move on to take another test. if not, it automatically diverges. the next tests are:
geometric, p-series, or integral teset.
nowwww, i know geometric is when you have an exponent, and p-series is when you haev a fraction. that's about all i know. therefore idk how to choose one, what to do after i choose one, or anything like that. i'm just so lost, idk why. i've been having a bad past few weeks.
anything else that i didn't mention in here, please explain to me. i think i need a tutor for this chapter or something.. idk.
I DO NOT UNDERSTAND:
direct comparison test
limit comparison test
^^i completely failed that quiz. ha
i'll just go over what i know.
so if you are given something, (idk if it matters if it is a sequence or a series).. you automaticaly take the nth term test.
so take lim as n goes to infinty. if you get 0, you have to move on to take another test. if not, it automatically diverges. the next tests are:
geometric, p-series, or integral teset.
nowwww, i know geometric is when you have an exponent, and p-series is when you haev a fraction. that's about all i know. therefore idk how to choose one, what to do after i choose one, or anything like that. i'm just so lost, idk why. i've been having a bad past few weeks.
anything else that i didn't mention in here, please explain to me. i think i need a tutor for this chapter or something.. idk.
I DO NOT UNDERSTAND:
direct comparison test
limit comparison test
^^i completely failed that quiz. ha
Ryan - October 24 Post
To be honest, I really have been lost since we started this chapter. It's probably because I didn't do my homework the first night, and I've never been able to catch up. So I'll start where I started to get lost (9.1).
*Sequence - a list of numbers.
*To find the terms in a sequence, simply plug in n (the term you are on) for x and solve.
*Sequences converge if they have a limit.
*Sequences diverge if they don't have a limit.
*To determine if a sequnce has a limit, take the limit as n->infinity of a(sub)n.
******Something we need to know!!!!!!!!
Limit(as n goes to infinity) of (1+(1/n))^n = e
*Sequence properties follow limit properties.
**Sqeeze Theorem.
- <= an <= +
*For sequences - sqeeze with a convergent sequence related to a(sub)n.
*Monotonic - if terms are always increasing or always decreasing.
*If a sequence is:
*bounded and monotonic - it converges
*bounded and not monotonic - it diverges
*not bounded and monotonic - it diverges
*Sequence - a list of numbers.
*To find the terms in a sequence, simply plug in n (the term you are on) for x and solve.
*Sequences converge if they have a limit.
*Sequences diverge if they don't have a limit.
*To determine if a sequnce has a limit, take the limit as n->infinity of a(sub)n.
******Something we need to know!!!!!!!!
Limit(as n goes to infinity) of (1+(1/n))^n = e
*Sequence properties follow limit properties.
**Sqeeze Theorem.
- <= an <= +
*For sequences - sqeeze with a convergent sequence related to a(sub)n.
*Monotonic - if terms are always increasing or always decreasing.
*If a sequence is:
*bounded and monotonic - it converges
*bounded and not monotonic - it diverges
*not bounded and monotonic - it diverges
Post #9
So, blogger is finally accepting my password and I can now do my blog the right way! Yayy!
So, this week, on the days I was here, we reviewed and took some quizzes.
I will explain, to the best of my knowledge of how to tell if something converges or diverges.
So, first things first you shoud check if it's geometric. Geometric is when something is being multiplied to every term in the series. If the thing that is getting multiplied to is less than one, the series CONVERGES.
Next, is alot like geometric..P-Series. Its p-series if you have a fraction and the bottom is n raised to the number exponent. If your number is less than or equal to one it DIVERGES, if its greater than one it CONVERGES.
Now, lets go over the limit comparison test, you take an equaiton and simplify it by taking the greatest exponent terms and taking the limit of that...
HERE COMES MY QUESTION: WHAT DOES THIS TELL YOUUUUUU?
I have the same question for Direct Comparison Test along with the integral test.
Can anyone tell me how these three tests give you an answer..and how do you know if it doesn't help you?
So, this week, on the days I was here, we reviewed and took some quizzes.
I will explain, to the best of my knowledge of how to tell if something converges or diverges.
So, first things first you shoud check if it's geometric. Geometric is when something is being multiplied to every term in the series. If the thing that is getting multiplied to is less than one, the series CONVERGES.
Next, is alot like geometric..P-Series. Its p-series if you have a fraction and the bottom is n raised to the number exponent. If your number is less than or equal to one it DIVERGES, if its greater than one it CONVERGES.
Now, lets go over the limit comparison test, you take an equaiton and simplify it by taking the greatest exponent terms and taking the limit of that...
HERE COMES MY QUESTION: WHAT DOES THIS TELL YOUUUUUU?
I have the same question for Direct Comparison Test along with the integral test.
Can anyone tell me how these three tests give you an answer..and how do you know if it doesn't help you?
Sunday, October 17, 2010
post 8
this week we learned more about convergence, divergence, p-series, nth term, integral test, and we learned something new called direct comparison test.
i'll be completely honest, and say that i am really confused with a lot of this stuff. idk why it just isn't sticking in my head.
i need someone to go over pretty much all of it... especially direct comparison test.
i also don't know the difference. whenever brob says what kind is this. idk how to tell if it's p-series, or something else. i'm really lost. idk, this just wasn't my best week.
i know they all have to do with sigmas. and i know what to do if the steps are in front of me. but idk, i guess i'm just lost.
sorry if this is a short post, i'm just in need of help mostly.
i'll go over at least one thing.
INFINITY RULES:
top degree > bottom degree = +/- infinity.
top degree < bottom degree = 0
top degree = bottom degree = divide coefficients
these are actually very helpful and never go away. a lot of the time we just automatically do l'hospitals rule whenever we could just be doing this!
i'll be completely honest, and say that i am really confused with a lot of this stuff. idk why it just isn't sticking in my head.
i need someone to go over pretty much all of it... especially direct comparison test.
i also don't know the difference. whenever brob says what kind is this. idk how to tell if it's p-series, or something else. i'm really lost. idk, this just wasn't my best week.
i know they all have to do with sigmas. and i know what to do if the steps are in front of me. but idk, i guess i'm just lost.
sorry if this is a short post, i'm just in need of help mostly.
i'll go over at least one thing.
INFINITY RULES:
top degree > bottom degree = +/- infinity.
top degree < bottom degree = 0
top degree = bottom degree = divide coefficients
these are actually very helpful and never go away. a lot of the time we just automatically do l'hospitals rule whenever we could just be doing this!
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