Diverge vs. Converge

Jan 2010
72
2
Does this series converge or diverge?

[(2k+1)^(2k)]/[(5k^2+1)^k]
 
Dec 2009
872
381
1111
Does this series converge or diverge?

[(2k+1)^(2k)]/[(5k^2+1)^k]
Dear WartonMorton,

\(\displaystyle \frac{(2k+1)^{2k}}{(5k^2+1)^k}=\left(\frac{(2k+1)^2}{5k^2+1}\right)^k\)

Now divide \(\displaystyle (2k+1)^2~and~5k^2+1~by~k^2\)

Hope you can continue from here.
 
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Jan 2010
72
2
Dear WartonMorton,

\(\displaystyle \frac{(2k+1)^{2k}}{(5k^2+1)^k}=\left(\frac{(2k+1)^2}{5k^2+1}\right)^k\)

Now divide \(\displaystyle (2k+1)^2~and~5k^2+1~by~k^2\)

Hope you can continue from here.

Once I do that I am seeing convergence?
 
Dec 2009
872
381
1111
Once I do that I am seeing convergence?
Dear WartonMorton,

Is it so? The term inside the bracket converges to 2/5 (when k tends to infinity). But the whole term has a \(\displaystyle k^{th}\) power. So.....