Convergence of sequence using definition

"using the definition (epsilon-N definition), show that the sequences (-1)^n(1/3n) + (1/2n) and cos(n^2)/(3n+1) both converge to the limit zero". ok, so i know the limit and i know the process. i can handle these questions ok when it is just a polynomial sequence but, for example, what do i do with the (-1)^n? the first step i need to do is deal with this and i don't know how to proceed. similar for cosines.

Re: Convergence of sequence using definition

Can you show us what you tried please?

Re: Convergence of sequence using definition

hi,

as i said, i encountered a problem with the first step. if someone can tell me how to deal with the (-1)^n, i can do the rest. but i cant try anything beyond that because the obstacle i need help with is right at the beginning..

Re: Convergence of sequence using definition

Quote:

Originally Posted by

**learning** hi,

as i said, i encountered a problem with the first step. if someone can tell me how to deal with the (-1)^n, i can do the rest. but i cant try anything beyond that because the obstacle i need help with is right at the beginning..

For the first one separate it.

Make each of those terms less than .

Re: Convergence of sequence using definition

hmmm its 1/3n, not (1/3)^n. so does this make a difference? my thought was that i should just say that the whole thing is less than (something?) regardless of sign so i.e. take the value for even n and say its less than that?

Re: Convergence of sequence using definition

To prove , i.e

Proof:

Let then we have

.

Since .

Can you finish the proof?

Re: Convergence of sequence using definition

what i did is:

etc.... is this the same/an ok way of expressing it? basically saying that 'it has to be less than what it is if n is even', as that gives the highest absolute value. i mean its technically true, do i have to be more rigid in the conditions here? it leads to same the resut. and yes i am confortable with the rest of the proof it was just how to deal with the (-1)^n i have never come across. basically the differnce is i put the two parts over a common denomnator instead of splitting. in fact once the (-1)^n is gone i can just get rid of the modulus altogether since its all always positive then.

Re: Convergence of sequence using definition

Can you give the proof for the other limit now?