Step by step solution to "Show that the following sequence converges an=13(1−13n)"

Kasey Haley

Answered question

2022-01-24

Show that the following sequence converges

a_{n} = \frac{1}{3} (1 - \frac{1}{3^{n}})

Answer & Explanation

Jaiden Conrad

Beginner2022-01-25Added 14 answers

a_{n} = \frac{1}{3} (1 - \frac{1}{3^{n}})

Itis known that a monotonic decreasing sequence bounded is convergent and it converges to the greatest lower bound and a monotonic increasing sequence bounded above converges to the least upper bound.
For the given sequence

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} \times \frac{1}{3^{n}} > 0

Since,

a_{n + 1} > a_{n}

, therefore the sequence is monotonic increasing.
Now,
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}

Hence, being monotonic increasing sequence bounded above,

\frac{1}{3}

being upper bound, the given sequence

a_{n}

is convergent

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