Limit of a sequence

9000064003

Level: 
C
Consider the convergent sequence \[ (a_{n})_{n=1}^{\infty } = \left (\frac{4n^{2} + 3n - 250} {2n^{2}} \right )_{n=1}^{\infty } \] and its limit \(L\). Find the maximal difference between \(L\) and the subsequence \((a_{n})_{n=250}^{\infty }\). (In other words, find the maximal difference between \(L\) and the terms of the sequence starting at \(a_{250}\).)
\(0.004\)
\(0.04\)
\(0.504\)
\(0.54\)

9000064008

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

9000064009

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

9000064010

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