Question

using definition of a null sequence prove that a_n=(1+(-1)^n)/n+(-1)^n, n=2,3,... is null

Answer 1

\[\lim _{n \rightarrow \infty} a _{n}=\lim _{n \rightarrow \infty} (1+(-1)^{n})/n+(-1)^{n}\]
clearly the numerator is 2,0,2,0,2,0,2,0.......
and the denominator is growing infinitly. So the a_n is getting smaller and smaller.
By definition \[|a _{n}-0| =|a _{n}|<\epsilon \] since lim a_n is 0

Answer 2

no

Answer 3

use what i wrote befor

Answer 4

by definition, you have to find n big enough so that for this n
\[|a _{n}-0|=|a _{n}|<\epsilon\]

Answer 5

\[|1+(-1)^{n}/n+(-1)^{n}|<\]

Answer 6

epsilon missing

Answer 7

for odd n |a_n| is 0. so just do it for even n:
2/n+1<e
2/e-1<n

Answer 8

ok?

Answer 9

got it?

Answer 10

yw

Answer 11

you wellcome

Answer 12

ah sorry new to all this, thanks