How do I calculate the following limit: $\displaystyle\lim_{k \rightarrow \infty } \sqrt[k]{k(k+1)}$
The only limit identity that I know which closely resembles this is $\displaystyle\lim_{k \rightarrow \infty } \sqrt[k]{k}=1$.
Edit: This question came in context of finding the radius of convergence of $\displaystyle\sum_{k=1}^\infty \dfrac{2^k z^{2k}}{k^2+k}$.
Attempt: Clearly, the ratio test here is inconclusive. So, I tried Cauchy-Hadamard Formula.
For the general form of a power series, this says that $R=\displaystyle \dfrac{1}{\limsup_{k \rightarrow \infty} \sqrt[k]{|a_k|}}$.
But now $\displaystyle \lim_{k \rightarrow \infty} \sqrt[k]{\dfrac{2^k}{k^2+k}}=2 \lim_{k \rightarrow \infty} \sqrt[k]{\dfrac{1}{k^2+k}}=2$ as per the calculations done by @gimusi, @Key Flex.
Doubt: The answer at the back of the book gives $R=\frac{1}{\sqrt{2}}$.
Best Answer
We have that
$$ \sqrt[k]{k(k+1)}= \sqrt[k]{k} \,\sqrt[k]{k+1} \to 1 \cdot 1=1$$
indeed
$$\sqrt[k]{k+1}=e^{\frac{\ln k}{k+1}}\to e^0=1$$
and more in general for any polynomial $p_n(k)$ we have
$$\sqrt[k]{p_n(k)} \to 1$$
by the same proof.