Convergent Sequence: t^n/(n Factorial) Limit as n->∞?

[aroosak]

i have a sequence an= t^n / (n factorial).
I know that the infinite series of it converges to zero, but i need to know if the limit of an goes to zero or not , as n goes to infinity.

Thanks

 

[fourier jr]

i have a sequence an= t^n / (n factorial).
I know that the infinite series of it converges to zero,

are you sure about that? I think the series would converge to something like $$e^t$$, that is if you mean the series $$\sum_{0 \leq n} \frac{t^n}{n!}$$

but i need to know if the limit of an goes to zero or not , as n goes to infinity.

Thanks

if a series converges as you say, then its terms necessarily tend to 0

 

[Landau]

You probably mixed the things up: you know that a_n->0 as n->\infty (which is a necessary but not sufficient condition for the series to converge), and you wonder whether the series converges.

 

[aroosak]

yes... fo course if converges to the exponential function.

and i just found a theorem, saying that it makes each term to vanish as well.

thank you though.. i think sometimes i get confuesdif i spend too much time on one topic.

 

[aroosak]

there is one more thing...
what if i start the sum from 1 or even some random constant k, will the sum of an= t^n / (n factorial) still go to the exponential function as n goes to infinity? i mean if we just consider the tail of the sequence, will the series still go to e^t?

thank you

 

[mathman]

there is one more thing...
what if i start the sum from 1 or even some random constant k, will the sum of an= t^n / (n factorial) still go to the exponential function as n goes to infinity? i mean if we just consider the tail of the sequence, will the series still go to e^t?

thank you

The series will will sum to e^t - all the terms you left out.

 

[aroosak]

thanks very much