Question

I’m having trouble I need this answered, it is apart of my ACT prep guide

Answer 1

Question:

Solution:

According to the data of the problem, the series is given by the following expression:

$\sum ^{\infty}_{n\mathop=1}\frac{n}{3^n}=\frac{1}{3^1}+\frac{2}{3^2}+\frac{3}{3^3}+\cdots$

now, remember the ratio test:

Suppose we have the series

$\sum ^{}_{}a_n$

Define,

$L\text{ =}\lim _{n\to\infty}|\frac{a_{n+1}}{a_n}|$

Then,

if L<1, the series is absolutely convergent (and hence convergent).

if

L>1, the series is divergent.

if

L=1 the series may be divergent, conditionally convergent, or absolutely convergent.

Applying this definition to the given series, we obtain:

$L\text{ =}\lim _{n\to\infty}|\frac{(n+1)3^n_{}}{n3^{n+1}_{}}|=\frac{1}{3}<1$

then, the given series is absolutely convergent (and hence convergent). So that, we can conclude that the correct answer is: