Does the series k a^k have a name?

[zeroseven]

Hi, a naive question here, but I was wondering if the series

$\sum^{∞}_{k=0}$ k a$^{k}$

has a particular name? As in 'geometric series' for $\sum^{∞}_{k=0}$ a$^{k}$ ?

And what about the more general $\sum^{∞}_{k=0}$ k$^{n}$ a$^{k}$ ?As a related question, you seem to be able to get the formula for the sum by differentiating the formula for the geometric series. Is this a correct and mathematically rigorous way to do it?

Cheers,
zeroseven

 

[tiny-tim]

hi zeroseven!

no, no names so far as i know

As a related question, you seem to be able to get the formula for the sum by differentiating the formula for the geometric series. Is this a correct and mathematically rigorous way to do it?

yes, so long as both limits exist, d/dx ∑ f_n(x) = ∑ d/dx f_n(x) is rigorous

(can you prove it? )

 

[zeroseven]

Thanks tiny-tim!
I thought that might be the case. So many other series seem to have been named that I just thought it was odd!

No, I can't prove the identity you gave in your post... yet! But I will try.