really strange but nice question

i ran into this yesterday..
somebody asked me to help her on it
and the whole thing is quite strange and i cant see how i can explain it.

so the question is about 'infinite summation'

sum the sequece t_n

t_n = \dfrac{(xlna)^n}{n!}

a, x are variable. (n is from zero to infinity)

now the question asks you to vary a and x.
a=2, x=1, or a=3, x=1 and so on.

bottom line: the general statement which i got out is
as n

\rightarrow \infty S_n \rightarrow a^x

can somebody help me try to explain why?

Reply 1

negotiator90

i kind of tried to somehow use the geometric series but its kind of no use..

Reply 2

Totally Tom

consider power series for e^x

Reply 3

negotiator90

Totally Tom

consider power series for e^x

i kind of see what you're trying to point out
but i dont really see the whole thing..

\dfrac{(xlna)^n}{n!}

becomes

e^x * \dfrac{((lna)^n}{n!}

right?

how do go further?

Reply 4

Totally Tom

negotiator90

i kind of see what you're trying to point out
but i dont really see the whole thing..

\dfrac{(xlna)^n}{n!}

becomes e^x *

\dfrac{(na^n}{n!}

right?

how do go further?

wat.

try e^y with y=xloga...

Reply 5

negotiator90

Totally Tom

wat.

try e^y with y=xloga...

sorry for that. not used to letex

what i meant was

\displaystyle\sum_{n=0}^{\infty} \dfrac{(xlna)^n}{n!} = e^x *\displaystyle\sum_{n=0}^{\infty} \dfrac{(lna)^n}{n!}

was this what you were pointing out to?

Reply 6

Totally Tom

negotiator90

sorry for that. not used to letex

what i meant was

\displaystyle\sum_{n=0}^{\infty} \dfrac{(xlna)^n}{n!} = e^x *\displaystyle\sum_{n=0}^{\infty} \dfrac{(lna)^n}{n!}

was this what you were pointing out to?

no.

Reply 7

negotiator90

Totally Tom

no.

!!!!!!!!!!!!!!!!!!!!!!!!!

right i see what you mean..

the whole sequence is e^y , y=xlna
in form of the power series..

therefore e^xlna = a^x

so easy now..why did i miss it? :confused: