Question

$\lim _ { n \rightarrow \infty }$ $\left[ \left( 1 + \frac { 1 } { n } \right) \left( 1 + \frac { 2 } { n } \right) \left( 1 + \frac { 3 } { n } \right) \ldots \left( 1 + \frac { n } { n } \right) \right] ^ { 1 / n }$ =

• A. $\frac { \mathrm { e } } { 4 }$
• B. $\frac { 4 } { \mathrm { e } }$
• C. $\frac { 2 } { \mathrm { e } }$
• D. None of these

Answer 1

Answer: (B)

$\frac { 4 } { \mathrm { e } }$

Answer 2

A = $\lim _ { n \rightarrow \infty } \left[ \left( 1 + \frac { 1 } { n } \right) \left( 1 + \frac { 2 } { n } \right) \ldots . \left( 1 + \frac { n } { n } \right) \right] ^ { \frac { 1 } { n } }$

log A =

= = – $\int _ { 0 } ^ { 1 } \frac { 1 } { x + 1 } x d x$

= log2 – = log 2 –

= log 2 – [1 – log 2] = 2 log 2 – 1

log A = log 4 – 1

log A = log (4/e) ̃ A =