Is it possible to evaluate #sum_(n=1)^oosqrt(4n^2x^2-1)/(4n^2)# in terms of #x#?

Answer 1
For #x!=0# let:
#a_n = sqrt(4n^2x^2-1)/(4n^2)#

and:

#b_n = 1/(2nabsx)#

Now:

#lim_(n->oo) a_n/b_n = lim_(n->oo) 2nabsx sqrt(4n^2x^2-1)/(4n^2)#
#lim_(n->oo) a_n/b_n = x lim_(n->oo) sqrt(4n^2x^2-1)/(2n)#
#lim_(n->oo) a_n/b_n = absx lim_(n->oo) sqrt((4n^2x^2-1)/(4n^2))#
#lim_(n->oo) a_n/b_n = absx lim_(n->oo) sqrt(x^2-1/(4n^2))#
#lim_(n->oo) a_n/b_n = x^2 #
Based on the limit comparison test, for any #x!=0# the limit is finite and non null, so that the series must have the same characteristic.

But:

#sum_(n=0)^oo 1/(2nabsx) = 1/(2absx)sum_(n=0)^oo1/n#

is divergent, so also:

#sum_(n=0)^oo sqrt(4n^2x^2-1)/(4n^2)#

is divergent.

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Answer 2

Yes, it is possible to evaluate ( \sum_{n=1}^{\infty} \frac{\sqrt{4n^2x^2 - 1}}{4n^2} ) in terms of ( x ). This series is a convergent series, and it can be expressed as a function of ( x ).

The series can be simplified using techniques such as algebraic manipulation and the comparison test. By manipulating the terms of the series and comparing it to known convergent series, one can determine the convergence of the series and its value in terms of ( x ).

While the process of evaluating the series may involve complex mathematical techniques, it is indeed possible to express the sum in terms of ( x ) given that it converges for certain values 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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