# Summation Verification

Hi,

I'm just trying to evaluate a series and would just appreciate if someone could either verify or correct me work.

Essentially, I have a series that I've produced:

-[(t^2)/2 + (t^5)/(2x5) + (t^8)/(2x5x8) + ...]

= - *sum from n = 0 to infinity* [(t^(3n+2))/(3n+2)!] = -e^t

Sorry for the poor syntax. I'm just in a slight rush here and can hopefully fix it up sooner rather than later. Regardless, i essentially get -e^t as my answer, and if someone could verify if this is correct, that would be very helpful. Thank you!

Mark44
Mentor
Hi,

I'm just trying to evaluate a series and would just appreciate if someone could either verify or correct me work.

Essentially, I have a series that I've produced:

-[(t^2)/2 + (t^5)/(2x5) + (t^8)/(2x5x8) + ...]

= - *sum from n = 0 to infinity* [(t^(3n+2))/(3n+2)!] = -e^t
???
How do you figure that your series equals -et?
The Maclaurin series for -et is -(1 + t + t2/2! + ... + tn/n! + ...)
MathewsMD said:
Sorry for the poor syntax. I'm just in a slight rush here and can hopefully fix it up sooner rather than later. Regardless, i essentially get -e^t as my answer, and if someone could verify if this is correct, that would be very helpful. Thank you!

???
How do you figure that your series equals -et?
The Maclaurin series for -et is -(1 + t + t2/2! + ... + tn/n! + ...)
Yes, it seems very odd and certainly not equivalent. I tried simplifying the series, but I only got to -t2 sum [t^3n/(3n+2)! But I can't quite further simplify it from here to put it in a form that seems expressible in terms of a function of e^rt. Any hints? Is there any way for me to possibly simplify the factorial in the denominator? If I change the summation from n = 0 to n = 1 (and still to infinity), this would allow it to become t^(3n - 3)/(3n - 1)!, but that still doesn't really help....

Mark44
Mentor
What's the problem you're trying to solve?

What's the problem you're trying to solve?
There's actually no specific problem...

I essentially have this series (that was found in another problem) and although there are other representations which could help me solve this, i was wondering if there was a way for me to transform this into a function of e^rt....I don't quite see any ways to do so, but would welcome any suggestions or methods. I don't quite see how having t^(3n) terms in the sum can allow for the sum to be expressed as a function of e^rt.
I was also wondering if there are other ways to manipulate the denominator factorial to help make the sum become another expression.

lurflurf
Homework Helper
It is easy if you know about complex numbers
let
$$A=\sum_{k=0}^\infty \frac{x^{3k}}{(3k)!} \\ B=\sum_{k=0}^\infty \frac{x^{3k+1}}{(3k+1)!} \\ C=\sum_{k=0}^\infty \frac{x^{3k+2}}{(3k+2)!} \\ t=-\tfrac{1}{2}(1+i\sqrt{3}) \text{then solve for C using}\\ e^x=A+B+C e^{t x}=A+t B+t^2 C \\e^{t^2 x}=A+t^2 B+t C$$