Find the limit of the sequence of partial sums whose general term is an=10^n/n!
a) 10
b) 0 ----> my answer. Can you check for me, please?
c) 1
d) Does not exist
Note: I am between b and d answers
an -> 0
but you want the partial sums.
Think of the Taylor Series for e^x
To find the limit of the sequence of partial sums, let's first understand the terms of the sequence. We have a general term given by an = 10^n/n!.
Let's start by calculating some values of the sequence to observe any patterns:
For n = 1: a1 = 10^1/1! = 10/1 = 10
For n = 2: a2 = 10^2/2! = 100/2 = 50
For n = 3: a3 = 10^3/3! = 1000/6 = 166.67 (rounded to 2 decimal places)
For n = 4: a4 = 10^4/4! = 10000/24 = 416.67 (rounded to 2 decimal places)
As we calculate more terms of the sequence, we can observe that the terms become smaller and smaller.
To find the limit, let's examine the behavior of the sequence as n approaches infinity. We can calculate the limit using the formula:
lim(n→∞) an = lim(n→∞) 10^n/n!
We can rewrite 10^n as (10/1)^n and simplify:
lim(n→∞) an = lim(n→∞) (10/1)^n / n!
Using the rule for limits of exponentials, lim(n→∞) (10/1)^n = ∞, as 10/1 is greater than 1 and raised to increasingly large powers.
Next, let's consider the factor of n! in the denominator. n! grows much faster than (10/1)^n, so it dominates the behavior of the sequence.
As n approaches infinity, the denominator n! becomes extremely large, causing the value of the entire fraction to approach zero.
Therefore, the limit of the sequence of partial sums, lim(n→∞) an, is 0. Therefore, your answer (b) is correct.