1003047607
Level:
C
Find the limit \( \lim\limits_{n\to\infty} n\left( \sqrt n-\sqrt{n-1} \right) \).
\( \infty \)
\( \frac12 \)
\( 0 \)
\( 2 \)
\( -\infty \)
Limit of a sequence
1003047608
Level:
C
Choose the step to take first to efficiently evaluate the limit of the sequence \( \left( \frac{3n+2}{\sqrt{n^2-1}} \right)_{n=1}^{\infty} \).
We divide the numerator and the denominator by \( n \).
We take \( \sqrt n \) outside brackets in the numerator and the denominator.
We square the denominator.
We divide the numerator by \( n \).
We divide the denominator by \( n \).
1003047609
Level:
C
Find the limit \( \lim\limits_{n\to\infty}\frac{\sqrt{4n^2+3n+1}}{\sqrt{2n^2-5n-7}} \).
\( \sqrt2 \)
\( 2 \)
\( -\frac17 \)
\( 0 \)
\( \infty \)
1003047610
Level:
C
Find the limit \( \lim\limits_{n\to\infty} \frac{\sqrt{4n+5}}{\sqrt{2n^3+3n^2-1}} \).
\( 0 \)
\( \sqrt2 \)
\( 2 \)
\( \infty \)
\( -5 \)
1003047701
Level:
C
Find the limit \( \lim\limits_{n\to\infty}\frac{3+9+\dots+3^n}{4+16+\dots+4^n } \).
\( 0 \)
\( \frac32 \)
\( \infty \)
\( 1 \)
\( \frac34 \)
1003047702
Level:
C
Find the limit:
\[ \lim\limits_{n\to\infty}\left(\frac13+\frac19+\dots+\frac1{3^n} \right) \].
\( \frac12 \)
\( \frac13 \)
\( \frac32 \)
\( \infty \)
\( \frac23 \)
1003047703
Level:
C
Find the limit \( \lim\limits_{n\to\infty}\frac{\frac12+\frac14+\dots+\frac1{2^n}}{\frac13+\frac19+\dots+\frac1{3^n}} \).
\( 2 \)
\( \frac23 \)
\( \infty \)
\( 0 \)
\( \frac32 \)
1003047704
Level:
C
Find the limit. \[ \lim\limits_{n\to\infty}\frac{1^2+2^2+\dots+n^2}{n^2+7n-3} \]
Hint: \( 1^2+2^2+\cdots +n^2=\frac16 n(n+1)(2n+1) \).
\( \infty \)
\( 0 \)
\( \frac13 \)
\( -\frac13 \)
\( \frac16 \)
2010005301
Level:
C
Consider the sequence
\[
\left (2+\frac{(-1)^{n}}
{2n}\right )_{n=1}^{\infty }
\]
and its limit \(L\). How many terms
of the sequence differ from \(L\)
by more than \(\frac{1}
{100}\)?
\(49\)
\(50\)
\(10\)
\(100\)
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 \(a_{300}\).)
\(0.015\)
\(0.018\)
\(0.036\)
\(3.015\)