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.)
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Root test is simple Mal Pal. You just take the nth root the equation (the equation is the absolute value though). This will cancel your you exponent n. So all you have to do is take the limit as x goes to infinity which should be simple.
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