Answer
$ a.\quad$
Diverges.
See image. Steps given below.
$ b.\quad$
Diverges.
Work Step by Step
$a_{1}=1$
$a_{2}=1+(-2)=-1$
$a_{3}=1+(-2)+(-2)^{2}$
$a_{4}=1+(-2)+(-2)^{2}+(-2)^{3}$
$...$
$a_{n}=\displaystyle \sum_{k=1}^{n}(-2)^{n-1}$ = sum of the first n terms of a geometric sequence
$a_{n}= \displaystyle \frac{1-(-2)^{k}}{1-(-2)}=\frac{1-(-2)^{n}}{3}$
$a.\quad $
The steps you take will depend on the CAS you are using, but they follow the same logic.
Using the free online CAS at geogebra.org/cas:
Cell 1: Enter the function representing the sequence
$a(x)=\displaystyle \frac{5(1-5^{-x})}{4}$
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, the points to the right increase in magnitude and alternate above/below the x-axis.
The sequence seems to diverge.
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 " $?$ ".
(There is no limit)
$b.\quad $
The sequence diverges.
