Calculate Cn and Sn?

Question

#C_n=sumcosh(k x)#
#S_n= sum sinh(k x)#

where k=1 and n is an element of N*

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Answer 1 · 2017-11-28T10:53:53

See below.

#C_n = sum_(k=0)^n cosh(kx) = 1/2sum_(k=0)^n e^(kx)+1/2 sum_(k=0)^n e^(-kx)#

but, using the formula for the sum of geometric sequences,

#sum_(k=0)^n e^(kx) = (e^((n+1)x)-1)/(e^x-1)#

#sum_(k=0)^n e^(-kx) = (e^(-(n+1)x)-1)/(e^-x-1)#

and

#C_n = 1/2( (e^((n+1)x)-1)/(e^x-1)+ (e^(-(n+1)x)-1)/(e^-x-1)) = 1/2(1 + Cosh(n x) + Coth(x/2) Sinh(n x))#

with similar procedure we get at

#S_n = 1/2( (e^((n+1)x)-1)/(e^x-1)- (e^(-(n+1)x)-1)/(e^-x-1)) = Csch(x/2) Sinh((n x)/2)Sinh(1/2 (1 + n) x)#