How you solve this? #lim_(n-oo)prod_(k=1)^n((k+1)^2)/(k(k+2))#

Question

How you solve this?  #lim_(n->oo)prod_(k=1)^n((k+1)^2)/(k(k+2))#

Luke Alvarez · Accepted Answer

# lim_(n rarr oo) prod_(k=1)^n ((k+1)^2)/(k(k+2)) = 2#

We can find an exact formula for the product as follows: # prod_(k=1)^n ((k+1)^2)/(k(k+2)) # # " " = (1+1)^2/(1(1+2)) * (2+1)^2/(2(2+2)) * (3+1)^2/(3(3+2)) * ... * (n+1)^2/(n(n+2))# # " " = (2)^2/(1(3)) * (3)^2/(2(4)) * (4)^2/(3(5)) * ... * (n+1)^2/(n(n+2))# # " " = ( 2*3*4 * ... (n+1) )^2 / ((1*2*3 * ... * n)(3*4*5 * ... (n+2))) # # " " = (( 1*2*3*4 * ... (n+1) )^2 (1*2))/ ((1*2*3 * ... * n)(1*2*3*4*5 * ... (n+2))) # # " " = (2(n+1)!(n+1)!)/ ((n!)(n+2)!) # # " " = (2*n!(n+1)(n+1)!)/ ((n!)(n+1)!(n+2)) # # " " = (2(n+1))/(n+2) # And so: # lim_(n rarr oo) prod_(k=1)^n ((k+1)^2)/(k(k+2)) = lim_(n rarr oo) (2(n+1))/ ((n+2)) # # " " = lim_(n rarr oo) (2n+2)/(n+2) # # " " = lim_(n rarr oo) (2n+2)/(n+2) *(1/n)/(1/n)# # " " = lim_(n rarr oo) (2+2/n)/(1+2/n)# # " " = (2+0)/(1+0)# # " " = 2# As #lim_(n rarr oo) (2/n) =0 #

Ezra Adams · Answer

undefined To solve the limit lim_(n->oo)prod_(k=1)^n((k+1)^2)/(k(k+2)), we can simplify the expression inside the product notation and then evaluate the limit.

First, let's simplify the expression inside the product notation:

((k+1)^2)/(k(k+2)) = (k^2 + 2k + 1)/(k^2 + 2k) = (k^2 + 2k + 1)/(k(k + 2))

Now, we can rewrite the product notation as a fraction:

lim_(n->oo)prod_(k=1)^n((k+1)^2)/(k(k+2)) = lim_(n->oo)((2^2)/(1(1+2))) * ((3^2)/(2(2+2))) * ((4^2)/(3(3+2))) * ... * ((n^2)/((n-1)(n+1))) * (((n+1)^2)/(n(n+2)))

Next, we can simplify the expression further by canceling out common terms:

lim_(n->oo)prod_(k=1)^n((k+1)^2)/(k(k+2)) = lim_(n->oo)(2^2 * 3^2 * 4^2 * ... * n^2 * (n+1)^2) / (1 * 2 * 2 * 3 * 3 * 4 * ... * (n-1) * n * n * (n+1) * (n+2))

Now, we can see that many terms in the numerator and denominator cancel out:

lim_(n->oo)prod_(k=1)^n((k+1)^2)/(k(k+2)) = lim_(n->oo)(n+1)^2 / (n * (n+2))

Finally, we can evaluate the limit by dividing the highest power terms:

lim_(n->oo)prod_(k=1)^n((k+1)^2)/(k(k+2)) = 1

Therefore, the solution to the given limit is 1.

How you solve this? #lim_(n-&gt;oo)prod_(k=1)^n((k+1)^2)/(k(k+2))#

How you solve this? #lim_(n->oo)prod_(k=1)^n((k+1)^2)/(k(k+2))#