Concept:

Let, a_n is the n^th term of a sequence then if,

\(\mathop {\lim }\limits_{n \to ∞ } {a_n} = 0\)

then the sequence converges else the sequence diverges

Analysis:

\(X = \left\{ { - 1,\;1,\; - 1,\;1,\; \ldots } \right\}\)

a_n = (-1)ⁿ

<!--[if gte msEquation 12]>limn→∞an=-1∞=1≠0<![endif]--><!--[if !msEquation]--><!--[if gte vml 1]><![endif]--><!--[if !vml]--><!--[endif]--><!--[endif]-->\(\mathop {\lim }\limits_{n \to ∞ } {a_n} = {\left( { - 1} \right)^∞ } \ne 0\) 

∴ X is divergent

<!--[if gte vml 1]><![endif]--><!--[if !vml]-->\(Y = \left\{ {1,\frac{1}{2},\;3,\frac{1}{4}, \ldots } \right\}\)<!--[endif]--><!--[endif]-->

Now,

Y_n = n; if n = odd

= 1/n; if n = even

For even terms the sequence is convergent (Since 1/n tends to 0 as n tends to ∞ )

For odd terms the sequence is divergent. 

∴ We cannot say that the sequence is convergent.