Calculate Cn and Sn?

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

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

1 Answer
Nov 28, 2017

See below.

Explanation:

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)