Determine whether the nonincreasing sequence converges or diverges.
-$$a _ { n } = \frac { 1 + n \cdot 4 ^ { n } } { 4 ^ { n } }$$
A) Diverges
B) Converges

Question

Quizplus · Accepted Answer

We can use the ratio test to show that the series diverges:
\begin{align*}
\lim_{n 	o \infty} \frac{a_{n+1}}{a_n} &= \lim_{n 	o \infty} \frac{\frac{1+(n+1)4^{n+1}}{4^{n+1}}}{\frac{1+n\cdot 4^n}{4^n}} \
&= \lim_{n	o\infty}\frac{1+(n+1)4^{n+1}}{4^n(1+n\cdot 4^n)}\cdot\frac{4^n}{4^{n+1}}\
&= \lim_{n	o\infty}\frac{1+(n+1)4^{n+1}}{4(1+n\cdot 4^n)}\
&= \lim_{n	o\infty}\frac{4^{n+1}(n+1)+1}{4^{n+1}(4n+1)}\
&= \lim_{n	o\infty}\frac{4\cdot 4^{n}(n+1)+1}{4^{n+1}(4n+1)}\
&= \lim_{n	o\infty}\frac{4n+4+1/(4^n)}{4(4n+1)+1/(4^n)}\
&= \frac{4}{16} \
&= \frac{1}{4} < 1
\end{align*}
Since the ratio test tells us that the series diverges when the limit is less than 1, we conclude that the series diverges.

Determine Whether the Nonincreasing Sequence Converges or Diverges an=1+n⋅4n4na _ { n } = \frac { 1 + n \cdot 4 ^ { n } } { 4 ^ { n } }an​=4n1+n⋅4n​

Determine Whether the Nonincreasing Sequence Converges or Diverges $a _ { n } = \frac { 1 + n \cdot 4 ^ { n } } { 4 ^ { n } }$