Limit of a sequence

2010005304

Level: 
C
Find the limit of the following sequence. \[ {\left({\left( \root{n}\of{0.5}+\frac{\root{n}\of{0.5}} {n} \right)}^{n}\right)}_{ n=1}^{\infty } \] Hint: The limit of the sequence \({\left({\left(1 + \frac{1} {n}\right)}^{n}\right)}_{n=1}^{\infty }\) is the Euler number \(\mathrm{e}\).
\(\frac12 \mathrm{e}\)
\(\mathrm{e}^{\frac12}\)
\(\frac12 + \mathrm{e} \)
\(\infty \)

2010005303

Level: 
C
Find the limit of the following sequence. \[ {\left(\frac{(n^{2} + 4n + 4)^{n}} {n^{2n}} \right)}_{n=1}^{\infty } \] Hint: The limit of the sequence \({\left({\left(1 + \frac{2} {n}\right)}^{n}\right)}_{n=1}^{\infty }\) is \(\mathrm{e}^2\), where \(\mathrm{e}\) is the Euler number.
\(\mathrm{e}^{4}\)
\(\mathrm{e}+4\)
\(4\mathrm{e} \)
\(\infty \)

2010005302

Level: 
C
Consider the convergent sequence \[ (a_{n})_{n=1}^{\infty } = \left (\frac{6n^{2} + 10n - 300} {2n^{2}} \right )_{n=1}^{\infty } \] and its limit \(L\). Find the maximal difference between \(L\) and the subsequence \((a_{n})_{n=300}^{\infty }\). (In other words, find the maximal difference between \(L\) and the terms of the sequence starting at \(a_{300}\).)
\(0.015\)
\(0.018\)
\(0.036\)
\(3.015\)