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

• aroosak
In summary, the conversation discusses a sequence an= t^n / (n factorial) and the convergence of its infinite series. The speaker initially believes the series converges to zero, but the other person corrects them and says it converges to the exponential function. The speaker then asks if the limit of an goes to zero as n goes to infinity, and the other person confirms this but also mentions that it is not a sufficient condition for convergence. They also discuss starting the sum from different values and confirm that the series will still sum to et even if some terms are left out.
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

aroosak said:
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

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.

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.

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

aroosak said:
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 et - all the terms you left out.

thanks very much

## 1. What is a convergent sequence?

A convergent sequence is a sequence of numbers that approach a specific value as the sequence goes on. In other words, as the terms of the sequence increase, they get closer and closer to a certain number, known as the limit.

## 2. How do you find the limit of a convergent sequence?

To find the limit of a convergent sequence, you can use the formula lim(n->∞) t^n/(n factorial). This means that as n approaches infinity, the value of t^n/(n factorial) will approach a specific number, which is the limit of the sequence.

## 3. What is the significance of the n factorial in the convergent sequence formula?

The n factorial in the formula represents the product of all positive integers from 1 to n. It is important in the formula because it helps to control the growth rate of the sequence and ensures that the terms do not increase too quickly.

## 4. How does the value of t affect the limit of the convergent sequence?

The value of t can affect the limit of the convergent sequence in a few ways. If t is a positive number, the limit will also be a positive number. If t is negative, the limit will be negative. Additionally, if t is less than 1, the limit will be 0, and if t is greater than 1, the limit will be infinity.

## 5. Can you give an example of a convergent sequence using the formula t^n/(n factorial)?

One example of a convergent sequence using the formula t^n/(n factorial) is t^n/n! where t = 2. As n approaches infinity, the terms of this sequence will become increasingly smaller and approach 0 as the limit. This can be seen by plugging in larger and larger values for n, such as n = 10, 100, or 1000.

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