What is the radius of convergence of the MacLaurin series expansion for #f(x)= sinh x#?

Answer 1

#R=oo#

Let's first find the Maclaurin series expansion for #sinhx#:
#f(x)=sinhx=(e^x-e^-x)/2, f(0)=(e^0-e^0)/2=0#
#f'(x)=coshx=(e^x+e^-x)/2, f'(0)=(e^0+e^0)/2=1#
#f''(x)=sinhx, f''(0)=0#
#f'''(x)=coshx, f'''(0)=1#
#f^((4))(x)=sinhx, f^((4))(0)=0#
#f^((5))(x)=coshx, f^((5))(0)=1#

Thus, we observe a fairly regular pattern of ones and zeros alternately. Let's put the series' initial terms in writing:

The expansion of the Maclaurin Series is provided by

#f(x)=sum_(n=0)^oof^((n))(0)x^n/(n!)=f(0)+f'(0)x+f''(0)x^2/(2!)+...#

Thus, for our purpose, we obtain

#sinhx=0+x+0x^2+x^3/(3!)+0x^4+x^5/(5!)+...#

When we take out the terms that involve zero, we observe

#sinhx=x+x^3/(3!)+x^3/(3!)+...#
So, we want odd exponents and odd factorials starting at #1#, so the summation is
#sinhx=sum_(n=0)^oox^(2n+1)/((2n+1)!)#

We'll use the Ratio Test to determine the radius of convergence, where

#a_n=x^(2n+1)/((2n+1)!)#
#lim_(n->oo)|a_(n+1)/a_n|=lim_(n->oo)|x^(2n+3)/((2n+3)!)*((2n+1)!)/x^(2n+1)|#

Remove certain terms from the larger factorial because we want the factorials to cancel each other out:

#(2n+3)! = (2n+3)(2n+1)(2n+1)!#

Thus, we have

#lim_(n->oo)|(x^(2n+3)cancel((2n+1)!))/(x^(2n+1)(2n+3)(2n+2)cancel((2n+1)!))|#
#x^(2n+3)/x^(2n+1)=x^2#

So,

#|x^2|lim_(n->oo)1/((2n+3)(2n+2))<1# results in convergence.
The limit goes to #0.# Thus, this quantity is always #0<1# regardless of what we pick for #x#. We have convergence for all real numbers, IE, #R=oo#
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Answer 2

The radius of convergence of the Maclaurin series expansion for ( f(x) = \sinh(x) ) is infinite.

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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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