Reviewing real analysis (practice tests) and am asked to prove a sequence diverges. Proving a sequence converges is simple by using the definition, and I understand the divergence definition but am not sure how to utilize it to prove a sequence diverges.
I guess it depends on the limit but is typically the easiest way to prove a sequence is divergent by contradiction (assume it converges)? For example, I tried with this equation:
$$a_n = ln (n^2 – 2n)$$
to say "for contradiction, assume this sequence converges. Then:"
$$|ln(n^2+2n) – L| < \epsilon$$
but couldn't come up with anything conclusive/any way to explicitly depict n to get a contradiction. Any suggestions for this sequence and for proving divergence in general? Intro Real Analysis, so I know nothing will be too complicated my book just has no examples of this. Thanks for any help
Best Answer
Try this ; since you are doing real analysis it is enough to show the sequence is not Cauchy.
$|a_n-a_m|=|\ln(n^2-2n)-\ln(m^2-2m)|=|\ln(n)+\ln(n-2)-\ln m-\ln (m-2)|$
Now put $n=2m$
$|\ln 2+\ln m+\ln 2+\ln (m-2)-\ln m-\ln(m-2)|$
Cancelling we get
$\ln 2^2=2\ln 2$ which cant be made arbitrary less than any given $\epsilon >0$ which proves the claim