Answer
$ a.\quad$
See image. Steps given below.
$ b.\quad$
$N=652$
$N=116,678$
Work Step by Step
$ a.\quad$
The steps you take will depend on the CAS you are using, but they follow the same logic.
Here we use the free online CAS at "geogebra.org/cas".
Cell 1: Enter the function representing the sequence
$a(x)=x^{1/x}$
From the dropdown menu, select "Table of values".
In the dialog box for the table, set the range from 1 to 25, step 1.
When we observe the graph, it seems to be decreasing towards the value 1. The sequence seems to converge.
In the next free cell of the CAS, we find the limit when $ n\rightarrow\infty$. Here, we enter "L=Limit(a, infinity)" (without quotes).
The CAS returns the limit to be 1.
$ b.\quad$
We find the intersection of the graph of a(x) with the horizontal lines
$y=1.01$$\qquad $
Enter: Solve (a(x)=1.01)
We find:
$x=651.1$, so we take the next integer, $N=652.$
Check: $a(652)-1$ is under $0.01$.
Since $a$ is decreasing, all the following $a_{n} =a(n)$ terms are within 0.01 of the value of $L=1$.
and
$ y=1.0001$$\qquad $
Enter: Solve (a(x)=1.0001)
We find
$x=651.1$, so we take the next integer, $N=652$
Check: $a(116678)-1$ is under $0.0001$
Since $a$ is decreasing, all the following $a_{n} =a(n)$ terms are within 0.0001 of the value of $L=1$
