Explained: "Use what you know about bonded, monotonic sequences..."

Lennie Carroll

Answered question

2021-08-18

Use what you know about bonded, monotonic sequences to show that the following sequences converge
$a_{n} = \frac{1}{3} (1 - \frac{1}{3^{n}})$

Answer & Explanation

sweererlirumeX

Skilled2021-08-19Added 91 answers

For the given sequence ${a_{n}}$
$a_{n + 1} - a_{n} = \frac{1}{3} (1 - \frac{1}{3^{n + 1}}) - \frac{1}{3} (1 - \frac{1}{3^{n}})$
$= \frac{1}{3^{n}} \frac{1}{3^{n + 1}}$
$= \frac{1}{3^{n}} (1 - \frac{1}{3})$
$= \frac{2}{3} \frac{1}{3^{n}} > 0$
Since $a_{n} + 1 > a_{n}$ , therefore, the sequence ${a_{n}}$ is monotonic increasing.
For all $n \geq 1$
$(1 - \frac{1}{3^{n}}) < 1$
$\frac{1}{3} (1 - \frac{1}{3^{n}}) < \frac{1}{3}$
$a_{n} < \frac{1}{3}$ So, the sequence is bounded above with the upper bound $\frac{1}{3}$
Thus, being a monotonic increasing sequence bounded above, $\frac{1}{3}$ being an upper bound, the given sequence ${a_{n}}$ is convergent
It completes the proof.

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