Monday, 23 September 2013

calculus - Covergence of frac14+frac1cdot94cdot16+frac1cdot9cdot254cdot16cdot36+dotsb



I am investigating the convergence of the following series: 14+19416+192541636+1925364163664+



This is similar to the series 14+1346+135468+,which one can show is convergent by Raabe's test, as follows: an+1an=2n12n+2=2n+232n+2=132(n+1).



However, in the series I am looking at, an+1an=(2n1)2(2n)2=14n14n2, so Raabe's test doesn't work (this calculation also happens to show that the ratio and root tests will not work).




Any ideas for this one? It seems the only thing left is comparison...


Answer



I just read that there is a partial converse to Raabe's test that if an+1an1p/n for p1, then the series diverges. So I think that settles that it diverges.


No comments:

Post a Comment

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find lim without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...