# Diverge vs. Converge

#### WartonMorton

Does this series converge or diverge?

[(2k+1)^(2k)]/[(5k^2+1)^k]

#### Sudharaka

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.

WartonMorton

#### WartonMorton

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?

#### Sudharaka

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.....