MHB Determining whether the sequence converges or diverges as n--> infinity

[shamieh]

Can someone check my solutions?

Do the following sequences $${a_n}$$ converges or diverge as$$ \n\to\infty$$? If a sequence converges find its limit. Justify your answers.

1. $$a_n = 2 +(-1)^n$$

Answer: so can I say that as lim n --> infinity the sequence diverges by oscillation?

2. $$a_n = \frac{n}{e^n}$$

Answer: so: using l'opitals$$\frac{n}{e^n} = \frac{1}{e^n} = \frac{1}{\infty} = 0$$

$$\therefore$$ sequence converges to 0 ?

3. $$a_n = (1 + \frac{2}{n})^n$$

Answer: So using lopital I got $$(1)^n$$ so would that mean that the sequence diverges becase its 1 + 1 + 1 + 1 to infinity...

 

[Fermat1]

Can someone check my solutions?

Do the following sequences $${a_n}$$ converges or diverge as$$ \n\to\infty$$? If a sequence converges find its limit. Justify your answers.

1. $$a_n = 2 +(-1)^n$$

Answer: so can I say that as lim n --> infinity the sequence diverges by oscillation?

2. $$a_n = \frac{n}{e^n}$$

Answer: so: using l'opitals$$\frac{n}{e^n} = \frac{1}{e^n} = \frac{1}{\infty} = 0$$

$$\therefore$$ sequence converges to 0 ?

3. $$a_n = (1 + \frac{2}{n})^n$$

Answer: So using lopital I got $$(1)^n$$ so would that mean that the sequence diverges becase its 1 + 1 + 1 + 1 to infinity...

These are fine provided you do not have to prove from first principles

 

[shamieh]

so is my #3 correct? or will it converge to 1 since its a sequence?

 

[Fermat1]

so is my #3 correct? or will it converge to 1 since its a sequence?

I fear I've been writing rubbish. The series converges, in fact it is $e^2$. How you prove that depends on your definition of $e$