2000 Paper 3 Q7

Year: 2000
Paper: 3
Question Number: 7

Course: UFM Pure
Section: Sequences and series, recurrence and convergence

Difficulty: 1700.0 Banger: 1516.0

sequences convergence binomial theorem AM-GM inequality infinite product exponential function factorial inequality

Problem

Given that $$\e = 1 + {1 \over 1 !} + {1 \over 2 !} + {1 \over 3 !} + \cdots + {1 \over r !} + \cdots \; ,$$ use the binomial theorem to show that $$ {\left( 1 + {1 \over n} \right)}^{\!n} < \e $$ for any positive integer $n$. The product ${\rm P }( n )$ is defined, for any positive integer $n$, by $$ {\rm P} ( n ) = {3 \over 2} \cdot {5 \over 4} \cdot {9 \over 8} \cdot \ldots \cdot {2^n + 1 \over 2^n} . $$ Use the arithmetic-geometric mean inequality, $$ {a_1 + a_2 + \cdots + a_n \over n} \ge \ {\left( a_1 \cdot a_2 \cdot \ldots \cdot a_n \right)}^{1 \over n}\,, $$ to show that ${\rm P }( n ) < \e$ for all $n$ . Explain briefly why ${\rm P} ( n )$ tends to a limit as $n\to\infty$. Show that this limit, $L$, satisfies $2 < L\le\e$.

No solution available for this problem.

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Problem source

Given that 
$$\e = 1 + {1 \over 1 !} + {1 \over 2 !} + {1 \over 3 !} + \cdots + {1 \over r !} + \cdots \; ,$$ 
use the binomial theorem to show that 
$$ 
{\left( 1 + {1 \over n} \right)}^{\!n} < \e 
$$  
for any positive integer  $n$. 
 
The product ${\rm P }( n )$  is defined, for any positive integer $n$, by 
$$ 
{\rm P} ( n ) = {3 \over 2} \cdot {5 \over 4}  
\cdot {9 \over 8} \cdot \ldots \cdot {2^n + 1 \over 2^n} . 
$$ 
Use the arithmetic-geometric mean inequality, 
$$ 
{a_1 + a_2 + \cdots + a_n \over n}  
\ge 
 \ {\left( a_1 \cdot a_2 \cdot \ldots \cdot a_n \right)}^{1 \over n}\,, 
$$  
to show that ${\rm P }( n ) < \e$  for all $n$ . 
 
 
Explain briefly why  
 ${\rm P} ( n )$  tends to a limit as $n\to\infty$. 
Show that this limit, $L$, satisfies $2 < L\le\e$.

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