(6 points) Select the FIRST correct reason why the given series converges. A. Convergent geometric series B. Convergent p series C. Comparison (or Limit Comparison) with a geometric or p series D. Alternating Series Test E. None of the above

C. Comparison ( or limit of of comparison) with a geometric or p series.

Step-by-step explanation:

Let say we consider the series

S

=

∞

∑

n

=

1

a

n

.

For brevity, lets that assume

a

n

≥

0.

A common technique or way use for determining the convergence or divergence of a series is the the comparison test is. In a summarize detail, if we are able to show that the terms of S are bounded below by the terms of divergent series then we know that S diverges as well. Also, if can be able to bound the terms of S above by the terms of a convergent series then S will also converges as well. We commonly compare series to the geometric or p-series. A geometric series is of the form

∞

∑

n

=

0

r

n

and is convergent if

|

r

|

<

1.

A p-series is of the form

∞

∑

n

=

0

1

n

p

and it is convergent if

p

>

1.

the graph in the attached figure

step-by-step explanation:

we have

using a graphing tool

the graph in the attached figure

the roots are ( x-intercepts) x=-0.5,x=1,x=3

the y-intercept is the point (0,3)