Knowing that #sinh(x)=(e^x-e^(-x))/2# and that the Maclauren series for #sinh(x)#is #sum_(n=0)^oo (x^(2n+1))/((2n+1)!)#.Using the fact thatthe Maclauren series for #sinh(x)#converges to#sinh(x)# to showthat #sinh(x)# is an odd function #sinh(-x)=-sinh(x)?

#sinh(-x)=-sinh(x)#

1 Answer
Noah G
Mar 27, 2018

From the question we know that

#sinhx = sum_(n = 0)^oo x^(2n + 1)/((2n +1)!)#

Since #x^(2n)# will always be positive when x is negative (because #x^(2n)# will give an even power), #x^(2n + 1)# will always be negative when x is negative. #(2n + 1)!# will always be positive due to the nature of the factorial. Therefore, #sinhx# will always be negative whenever #x# is negative, satisfying the definition of an odd function, #sinh(-x) = -sinhx#.

Hopefully this helps!

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