How Do You Determine if a Series Converges Absolutely or Conditionally?

[sun1234]

https://lh3.googleusercontent.com/z468bYHn_-Ufcp0M_XFbTgUtEL2qDzNUZSRgjU3Qz8ZcFj8pMg5k7Ip9i9EejHNPdBQc3EltXrAI2Ck=w1342-h587

 

[micromass]

Just because the $n$th term converges to $0$, doesn't mean the series converges. Unless you can explain better why it might imply this in this case.

 

[micromass]

And the $p$-test only works for positive series (a series whose terms are positive).

 

[sun1234]

That's what I think of. Also how do you know when to test for absolute converges and conditional converges? Thank you for trying to help.

 

[Mark44]

That's what I think of.
Also how do you know when to test for absolute converges and conditional converges?

Instead of answering that question, I think it would be a good idea for you to step back and take a closer look at the two tests you used, the p-series test and what you call the "nth term test."
As already stated, the p-series applies only to series consisting of positive terms. You also misused the other test that you used. What exactly does that test say?

 

[HallsofIvy]

That's what I think of. Also how do you know when to test for absolute converges and conditional converges? Thank you for trying to help.

If a series "converges absolutely" then there is no point in asking if it converges conditionally. So it would seem to make sense to first try to show that a series converges absolutely and only if it doesn't try to show that it converges conditionally.

One test you do not mention is the "alternating sequence test": if, for $a_n> 0$, $\lim_{n\to 0} a_n= 0$ then $\sum (-1)^n a_n$ converges.