Question

Consider the sequence {an} = {nrn}. Decide whether {an} converges for each value of r.

Answer 1

Answer:

a). converges

b). diverges

c). converges

Step-by-step explanation:

{ $a_n$ } = { $nr^n$ }

Using radio test,

L = $\lim_{n \rightarrow \infty} |\frac{a_n + 1}{a_n}|$

   = $\lim_{n \rightarrow \infty} |\frac{(n+1)r^{n+1}}{nr^n}|$

   = $\lim_{n \rightarrow \infty} |(1+\frac{1}{n})^r|$

   = $\lim_{n \rightarrow \infty} |r|$

   = |r|

Therefore, $a_n$ converges in |r| < 1

a). r = 1/5

    $\{ a_n \}= \{\frac{1}{5}, n \}$

This sequence is monotonically decreasing and bounded.

0 < $a_n$ < 1

Hence, { $a_n$ } converges.

b). r = 1

    { $a_n$ } = { n }

This sequence is monotonically increasing sequence which is not bounded.

Hence, { $a_n$ } diverges.

c). r = 1/6

$\{ a_n \}= \{\frac{1}{6}, n \}$

This sequence is monotonically decreasing and bounded.

0 < $a_n$ < 1

Hence, { $a_n$ } converges.

For  |r| < 1, the $a_n$ converges.