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)#