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How to prove that #{1/2^n}# is bounded series ?

Answer 1

Isabella Ackerman

#{1/2^n}# is bounded since #0<1/2^n leq1/2#.

For all natural number #n#,

(a) #0 < 2^n Rightarrow 0 < 1/2^n#

(b) #n geq 1 Rightarrow2^n geq 2^1 Rightarrow 1/2^n leq 1/2^1#

So, #0<1/2^n leq 1/2# for all natural number #n#

Hence, #{1/2^n}# is bounded.

I hope that this was clear.

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Answer 2

Elijah Abram

To prove that the series {1/2^n} is bounded, we need to show that its terms do not exceed a certain value. In this case, each term of the series is 1 divided by 2 raised to the power of n. Since the denominator increases as n increases, the terms of the series decrease. However, the terms are always positive.

To establish a bound, note that the first term of the series is 1 (when n = 1), and subsequent terms become smaller as n increases. The series will never go beyond the first term because all subsequent terms are fractions that are smaller than 1. Hence, the series is bounded above by 1.

Additionally, all terms of the series are positive, so the series is also bounded below by 0.

Therefore, the series {1/2^n} is bounded between 0 and 1, inclusive.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.