How to Find the Sum of a Series with ln and Factorials as Terms?

[thyrsta]

Homework Statement

I have a question involving the sum of a series. I have already worked out a series, and the equation for a term in the series is:

((ln a)^n)/n!

a is just a variable, and n is the position in the sequence

a variation is:

(ln a)*((ln a)^(n-1))/n!

How would I start writing an equation for the sum of the series?

The Attempt at a Solution

I tried using the variation and treating it as a geometric series, with (ln a) as the first term, but that didn't work, for obvious reasons, since the second part isn't exactly the difference/scalar

 

[rock.freak667]

Homework Statement

I have a question involving the sum of a series. I have already worked out a series, and the equation for a term in the series is:

((ln a)^n)/n!

a is just a variable, and n is the position in the sequence

a variation is:

(ln a)*((ln a)^(n-1))/n!

How would I start writing an equation for the sum of the series?

The Attempt at a Solution

I tried using the variation and treating it as a geometric series, with (ln a) as the first term, but that didn't work, for obvious reasons, since the second part isn't exactly the difference/scalar

For a minute let's make a quick substitution of x=lna

does the infinite sum of xⁿ/n! look familiar?

 

[thyrsta]

For a minute let's make a quick substitution of x=lna

does the infinite sum of xⁿ/n! look familiar?

Sorry, no. I haven't learned much more beyond simple geometric and arithmetic series

 

[rock.freak667]

Sorry, no. I haven't learned much more beyond simple geometric and arithmetic series

Have you learned the Taylor/Maclaurin series of e^x?

 

[thyrsta]

Have you learned the Taylor/Maclaurin series of e^x?

No I haven't

 

[rock.freak667]

No I haven't

In that case, without using the direct result of


$$e^x = \sum_{n=0} ^{\infty} \frac{x^n}{n!}$$


I am not sure how to get you a closed for solution.

 

[thyrsta]

In that case, without using the direct result of


$$e^x = \sum_{n=0} ^{\infty} \frac{x^n}{n!}$$


I am not sure how to get you a closed for solution.

so e^{ln a}=a

But is there any way to have an equation that would give you the sum of a given number of terms in the series?

 

[Dick]

so e^{ln a}=a

But is there any way to have an equation that would give you the sum of a given number of terms in the series?

I don't think so. It's not a geometric series.

 

[thyrsta]

I don't think so. It's not a geometric series.

Ok thanks a lot guys for your help