# Thread: Nature of the series

1. ## Nature of the series

Could someone tell me how I can tell if the following sequences {an} diverge or converge? And how to justify it?

2. ## Re: Nature of the series

Calmo

liman = 0 , lim b = -1 and lim c = 0

very easy to ustify.....

3. ## Re: Nature of the series

a)Convergent. It is an alternating series, $\displaystyle a_n \to 0$ as $\displaystyle n \to \infty$

b) Divergent $\displaystyle a_n \to -1$ as $\displaystyle n \to \infty$ , its like adding -1 infinite times

c) divergent but a bit harder to show, multiply numerator and denominator by the conjugate. consider the following...

$\displaystyle a_n \ = \ \frac{(\sqrt{n + 1} - \sqrt{n}) (\sqrt{n + 1} + \sqrt{n}) }{ \sqrt{n + 1} + \sqrt{n}}$

$\displaystyle a_n \ = \ \frac{1}{\sqrt{n + 1} + \sqrt{n}}$

Now compare this with $\displaystyle b_n \ = \ \frac{1}{3 \sqrt{n}}$ which diverges because the exponent on n is less than 1

$\displaystyle a_n > b_n$ for n > 0 so a_n diverges because every nth term of a_n is LARGER than every corresponding nth term of b_n