Power of Reciprocal
Contents |
Theorem
Let $r \in \Q_{>0}$ be a strictly positive rational number.
Let $\left \langle {x_n} \right \rangle$ be the sequence in $\R$ defined as $x_n = \dfrac 1 {n^r}$.
Then $\left \langle {x_n} \right \rangle$ is a null sequence.
Real Index
If $r \in \R_{>0}$ is a strictly positive real number, the same result applies.
However, the result is specifically stated for a rational index, as this definition is used in the course of derivation of the existence of a power to a real index.
Corollary
Let $\left \langle {x_n} \right \rangle$ be the sequence in $\R$ defined as $x_n = \dfrac 1 n$.
Then $\left \langle {x_n} \right \rangle$ is a null sequence.
Proof
Let $\epsilon > 0$.
We need to show that $\exists N \in \N: n > N \implies \left|{\dfrac 1 {n^r}}\right| < \epsilon$.
That is, that $n^r > 1 / \epsilon$.
Let us choose $N = \left({1/\epsilon}\right)^{1/r}$.
Then $\forall n > N: n^r > N^r = 1 / \epsilon$.
$\blacksquare$
but you have to prove that $n^r$ is strictly increasing
You can help ProofWiki by explaining it.
To discuss this point in more detail, feel free to use the talk page.
If you are able to explain it, then when you have done so you can remove this instance of {{Explain}} from the code.
If you would welcome a second opinion as to whether your explanation is correct, add a call to {{Proofread}} the page (see the proofread template for usage).
Proof of Corollary
$n = n^1$ from the definition of power and as $1 \in \Q_{>0}$ the result follows.
$\blacksquare$
Notes
This result and Power of a Number Less Than One are sometimes referred to as the basic null sequences.
Also see
Sources
- K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 4.6 \ (2)$
