Calculate Cn and Sn?

C_n=sumcosh(k x)Cn=∑cosh(kx)
S_n= sum sinh(k x)Sn=∑sinh(kx)

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

1 Answer
Cesareo R. · NickTheTurtle
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)Cn=n∑k=0cosh(kx)=12n∑k=0ekx+12n∑k=0e−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)n∑k=0ekx=e(n+1)x−1ex−1

sum_(k=0)^n e^(-kx) = (e^(-(n+1)x)-1)/(e^-x-1)n∑k=0e−kx=e−(n+1)x−1e−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))Cn=12(e(n+1)x−1ex−1+e−(n+1)x−1e−x−1)=12(1+cosh(nx)+coth(x2)sinh(nx))

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)Sn=12(e(n+1)x−1ex−1−e−(n+1)x−1e−x−1)=csch(x2)sinh(nx2)sinh(12(1+n)x)

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