The Sum of Cosh(x) Values from O to N

[YouAreHexed]

Sum Problem

My working so far:

I expanded the sequence of cosh(x) values (Cosh(0) + Cosh(1) + Cosh(2) + ... + Cosh(N)) into (1/2(e⁰ + e⁰ + e¹ + e^-1 + e² + e^-2 +...+ eⁿ + e^-n)).

I then split them into the positive and negitive power group, Let A=(1/2(e⁰ + e¹ + e² ....+ eⁿ) and let B=(1/2(e⁰ + e^-1 + e^-2 + ... + eⁿ), with the hope of recombining them when i work out their sums to find the total sum, and thus the sum of cosh(x)

The Sequence A, is summarized by An = 1/2*eⁿ and B is summarized by Bn = 1/2 * e^-n.

Following the geometric series formula(Sequence = a*rⁿ, series Sn= (a(rⁿ⁺¹ -1))/(r-1) ) I've worked out the series of A to be = 1/2 * (eⁿ⁺¹-1)/(e-1).

However for B, the series formula does not give a correct answer, instead wolfram is insisting (e^(-N) (e^(N+1)-1))/(e-1) is the sum.

I followed through and that sum for B does give the correct answer, at least in base e form, and i am unsure of how to reach this answer.

Any help you could give would be greatly appreciated, thanks in advance

 

[Steven G]

www.wolframalpha.com/input/?i=sum+cosh%28x%29+0+to+N

I need to show sum of Cosh(x) values from 0 to N, in terms of mainly cosh(N+1), sinh(N+1) and cosh(1) and 1,2 ect.

My working so far:

I expanded the sequence of cosh(x) values (Cosh(0) + Cosh(1) + Cosh(2) + ... + Cosh(N)) into (1/2(e⁰ + e⁰ + e¹ + e^-1 + e² + e^-2 +...+ eⁿ + e^-n)).

I then split them into the positive and negitive power group, Let A=(1/2(e⁰ + e¹ + e² ....+ eⁿ) and let B=(1/2(e⁰ + e^-1 + e^-2 + ... + eⁿ), with the hope of recombining them when i work out their sums to find the total sum, and thus the sum of cosh(x)

The Sequence A, is summarized by An = 1/2*eⁿ and B is summarized by Bn = 1/2 * e^-n.

Following the geometric series formula(Sequence = a*rⁿ, series Sn= (a(rⁿ⁺¹ -1))/(r-1) ) I've worked out the series of A to be = 1/2 * (eⁿ⁺¹-1)/(e-1).

However for B, the series formula does not give a correct answer, instead wolfram is insisting (e^(-N) (e^(N+1)-1))/(e-1) is the sum.

https://www.wolframalpha.com/input/?i=Sum+1%2Fx%5En+from+0+to+N

I followed through and that sum for B does give the correct answer, at least in base e form, and i am unsure of how to reach this answer.

Apart from that, how to convert the answer into hyperbolic function based answer is another problem i had as i am unsure how to change the answer into an answer 'mainly incorperating' Cosh(N+1), Sinh(N+1) ect.

Any help you could give would be greatly appreciated, thanks in advance

Hi, I am not sure where your trouble is, sorry. Here is what I get for B.
e^-0 + e^-1 + ... + e^-n = (1/e)^0 + (1/e)^1 + ... + (1/e)^n = (1- (1/e)^(n+1))/(1-1/e)= (e-(1/e)^n)/(e-1). Is this what you want?

 

[YouAreHexed]

Sorry Mate, i made a silly mistake. Sorry to waste your time and thanks for your help