Math Help - Determining if An is convergent or not???!?!??!?!????????? And explain WHY!!

1. Determining if An is convergent or not???!?!??!?!????????? And explain WHY!!

Let an is the following.

(a) Determine whether {an} is convergent.

(b) Determine whether is convergent.

Also the answer to be is that it is not convergent, but why????????

2. Re: Determining if An is convergent or not???!?!??!?!????????? And explain WHY!!

Originally Posted by yankees2694
Let an is the following.
(a) Determine whether {an} is convergent.
(b) Determine whether is convergent.

The part a) is convergent divide all by $n$.

For part b) $\sum\limits_{n = 1}^\infty {{a_n}}$ converges only if $(a_n)\to 0$.

3. Re: Determining if An is convergent or not???!?!??!?!????????? And explain WHY!!

a) Divide all parts of the fraction by the dominant term (the term that increases fastest when n increases). Then see what happens when n tends to infinity.

b) The infinite sum can only converge if an is a null sequence.

Haha Plato replied less than a minute before me.

4. Re: Determining if An is convergent or not???!?!??!?!????????? And explain WHY!!

So for (b), your series diverges since the terms do not approach zero (you will show in part (a) that they approach a different number). Some textbooks call this the Divergence Test.

I hope that clarifies Plato and Shakarri's answer a little.

- Hollywood

5. Re: Determining if An is convergent or not???!?!??!?!????????? And explain WHY!!

Yankey :

Lim(an) = 1/2 different than zero...therefore the Σan diverges...the divergent test as Holywood mentioned....