9000073406
Level:
A
Find the sum of the following infinite series.
\[
\sum _{n=1}^{\infty }\left (\frac{\sqrt{2} - 1}
{\sqrt{2}} \right )^{n-1}
\]
\(\sqrt{2}\)
\(\frac{\sqrt{2}+1}
{\sqrt{2}} \)
\(\frac{\sqrt{2}}
{2} \)
Series diverges.
Infinite Series
9000073401
Level:
B
In the following list identify the expression which equals
\(3.3\overline{12}\).
\(3.3 +\sum _{ n=1}^{\infty }12\cdot 10^{-2n-1}\)
\(3 +\sum _{ n=1}^{\infty }312\cdot 10^{-2n-1}\)
\(3 +\sum _{ n=1}^{\infty }312\cdot 10^{-3n}\)
\(3.3 +\sum _{ n=1}^{\infty }12\cdot 10^{-3n}\)
9000063403
Level:
A
Evaluate the following infinite product.
\[
2\cdot \sqrt{2}\cdot \root{4}\of{2}\cdot \root{8}\of{2}\cdot \cdots
\]
\(4\)
\(1\)
\(2\)
\(8\)
9000063404
Level:
A
Evaluate the following infinite sum.
\[
\frac{5}
{2} + \frac{5}
{8} + \frac{5}
{32} + \frac{5}
{128}+\cdots
\]
\(\frac{10}
{3} \)
\(5\)
\(4\)
\(\frac{5}
{2}\)
9000063410
Level:
B
Solve the following equation.
\[
x + \frac{x}
{3} + \frac{x}
{9} + \frac{x}
{27}+\cdots = 18
\]
\(x = 12\)
\(x = 6\)
\(x = 18\)
\(x = 24\)
9000063407
Level:
B
In the following list find the value of the parameter
\(x\) which
ensure that the following geometric series is divergent.
\[
\sum _{n=1}^{\infty }(x + 4)^{2n}
\]
\(x = -5\)
\(x = -\frac{9}
{2}\)
\(x = -4\)
\(x = -\frac{7}
{2}\)
9000063408
Level:
B
In the following list find the value of the parameter
\(x\) which
ensure that the following geometric series is divergent.
\[
\sum _{n=1}^{\infty }(5 - 3x)^{n}
\]
\(x = \frac{1}
{2}\)
\(x = \frac{13}
{9} \)
\(x = \frac{11}
{6} \)
\(x = \frac{5}
{3}\)
9000063401
Level:
A
Identify the quotient of the geometric series
\(\sum _{n=1}^{\infty } \frac{1}
{2^{n-3}} \).
\(\frac{1}
{2}\)
\(2\)
\(1\)
\(\frac{1}
{8}\)
9000063402
Level:
A
Identify the quotient of the geometric series
\(\sum _{n=1}^{\infty }3^{2-n}\).
\(\frac{1}
{3}\)
\(1\)
\(\frac{1}
{9}\)
\(-\frac{1}
{9}\)
9000063406
Level:
A
Evaluate the following infinite sum.
\[
\sum _{n=1}^{\infty }\left (-\frac{1}
{2}\right )^{n+2}
\]
\(- \frac{1}
{12}\)
\(-\frac{1}
{8}\)
\(\frac{1}
{2}\)
\(1\)