Post # 10

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!

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

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.)

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