Respuesta :
Answer:
a). converges
b). diverges
c). converges
Step-by-step explanation:
{ [tex]$ a_n $[/tex] } = { [tex]$ nr^n $[/tex] }
Using radio test,
L = [tex]$ \lim_{n \rightarrow \infty} |\frac{a_n + 1}{a_n}| $[/tex]
= [tex]$ \lim_{n \rightarrow \infty} |\frac{(n+1)r^{n+1}}{nr^n}| $[/tex]
= [tex]$ \lim_{n \rightarrow \infty} |(1+\frac{1}{n})^r| $[/tex]
= [tex]$ \lim_{n \rightarrow \infty} |r| $[/tex]
= |r|
Therefore, [tex]$a_n$[/tex] converges in |r| < 1
a). r = 1/5
[tex]$\{ a_n \}= \{\frac{1}{5}, n \}$[/tex]
This sequence is monotonically decreasing and bounded.
0 < [tex]$ a_n $[/tex] < 1
Hence, { [tex]$ a_n $[/tex] } converges.
b). r = 1
{ [tex]$ a_n $[/tex] } = { n }
This sequence is monotonically increasing sequence which is not bounded.
Hence, { [tex]$ a_n $[/tex] } diverges.
c). r = 1/6
[tex]$\{ a_n \}= \{\frac{1}{6}, n \}$[/tex]
This sequence is monotonically decreasing and bounded.
0 < [tex]$ a_n $[/tex] < 1
Hence, { [tex]$ a_n $[/tex] } converges.
For |r| < 1, the [tex]$a_n$[/tex] converges.
