How to calculate limit? $lim_(n->oo)sum_(k=0)^n(C_n^k)/(2n)^k$

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Answer 1 · 2017-03-27T16:06:14

$e^(1/2)$

$lim_(n->oo)sum_(k=0)^n(C_n^k)/(2n)^k$

$(1+y)^n =sum_(k=0)^n((k),(n))y^k$ now making

$y = x/(2n)$ we have

$(1+x/(2n))^n =sum_(k=0)^n((k),(n))(x/(2n))^k$

then

$lim_(n->oo)(1+x/(2n))^n=e^(x/2)$ Now making $x=1$

we have

$lim_(n->oo)(1+1/(2n))^n=e^(1/2)$

How to calculate limit?lim_(n->oo)sum_(k=0)^n(C_n^k)/(2n)^klim_(n->oo)sum_(k=0)^n(C_n^k)/(2n)^k