How to calculate this limit?#lim_(n->oo)prod_(k=1)^ncos(2^(k-1)x);x!=kpi#

Answer 1

#0#

#sin(2^kx)=2cos(2^(k-1)x)sin(2^(k-1)x)#

or

#f_k = 2 cos(2^(k-1)x)f_(k-1)# #f_(k-1) = 2 cos(2^(k-2)x)f_(k-2)# #cdots# #f_1 = 2 cos(x)f_0#

then

#f_k = (2^k prod_(j=0)^k cos(2^jx)) f_0#

then

# prod_(j=0)^n cos(2^jx)=1/2^n(f_n)/(f_0)=1/2^n(sin(2^n x)/sinx)#
we can then conclude that for #x ne k pi#
#lim_(n->oo)prod_(j=0)^n cos(2^jx)=0#
because if #x ne k pi# with #k in ZZ# we have
#abs(sin(2^n x)/sinx) le M#
so there exists #n# such that #M/2^n < epsilon# for arbitrarily small #epsilon#.
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Answer 2

To calculate the limit ( \lim_{n \to \infty} \prod_{k=1}^n \cos\left(2^{k-1}x\right) ) given that ( x ) is not equal to ( k\pi ) for any integer ( k ), we can utilize the following steps:

  1. Recognize that the given limit involves an infinite product of cosine functions.
  2. Note that the cosine function oscillates between -1 and 1 as its argument varies.
  3. Observe that when ( x ) is not equal to ( k\pi ), the argument of each cosine function in the product will never be a multiple of ( \pi ), ensuring that the cosine function remains bounded between -1 and 1.
  4. Understand that the product of cosine functions will converge to zero if any of the cosine terms approach zero as ( n ) tends to infinity.
  5. Realize that since ( x ) is not equal to ( k\pi ) for any integer ( k ), the cosine terms will never be zero for any ( n ), and thus, the product of cosine functions will converge to a nonzero value as ( n ) tends to infinity.
  6. Therefore, the limit ( \lim_{n \to \infty} \prod_{k=1}^n \cos\left(2^{k-1}x\right) ) converges to a nonzero value, but its specific value depends on the particular value of ( x ).
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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