Question

Determine the convergence or divergence. If the sequence converges, find the limit.

Answer 1

\[\left\{ an \right\} = \left\{ 1+\frac{ (-1)^{n} }{ n } \right\}\]

Answer 2

not sure how to go about starting this. would i let {an}=f(x) then try to solve the limit?

Answer 3

the sequence converges, why??
a0<a2< ... a3<a1

Answer 4

i know if it was just(-1)^n/n it would converge by the absolute theorm but, im not sure what the additional 1 would do.

Answer 5

because f(x) is not defined on [1, infinity)

Answer 6

...???

Answer 7

sorry ... i was busy :((

Answer 8

notice that is is squeezed between odd terms and even terms

Answer 9

so its the squeeze theorm?

Answer 10

why is it squeezed?

Answer 11

somewhat like that
\[ a_{2n}>a_{2n-2}> ..... >a_{2n-1}>a_{2n+1}\]

Answer 12

sorry ,, i put in opposite order
\[ a_{2n+2}>a_{2n}> ..... >a_{2n+1}>a_{2n-1} \]

Answer 13

again you need to concept of limit point. do you know monotone convergence theorem?

Answer 14

we haven't done that. we've done squeeze, absolute, test for convergence, nth term test for divergence, convergence of a geometric sequence, limits of functions, one with l'hopital rule.

Answer 15

just show that
\[ \lim_{n\to\infty } |1 + (-1)^n/n - 1| = 0\]
it is enough to prove the limit of sequence is 1.

Answer 16

so it is the absolute value theorem'

Answer 17

no ... definition of limit

Answer 18

where are you getting "n-1" from?

Answer 19

haa ... looks like you didn't get it