Tuesday, 12 June 2018

real analysis - divergence of sumin=3nftyfracsqrtn+2n2 verification/ alternative method



I wish to prove divergence of
n=3n+2n2




I wish to do so by comparison, since n3:
n=3n+2n2>n=31+2n2>n=33n>n=31n


And the harmonic series is divergent, so if we just remove finitely many terms, we still have that it is divergent, because divergence is determined "in the tail". We have a divergent minorant series and hence the original series diverges to .



Is this approach fine, or is there some more elegant method, this was about the simplest thing I could think of.






Alternatively we have:
n=3n+2n2>n=3n+2n=n=31n+2n



Answer



Not clear what your question is, but your answer is correct.



Your approach is fine. Comparison test would be the proper test to use.



One can also show that n=3n+2n2n=3n+2n



and show that the rightmost sum is diverging via the integral test.


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