Nth Root Test

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Theorem

Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be a series of real numbers $\R$ or complex numbers $\C$.

Let the sequence $\sequence {a_n}$ be such that the limit superior $\displaystyle \limsup_{n \mathop \to \infty} \size {a_n}^{1/n} = l$.

Then:

If $l > 1$, the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ diverges.
If $l < 1$, the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ converges absolutely.


Proof

Absolute Convergence

Let $l < 1$.

Then let us choose $\epsilon > 0$ such that $l + \epsilon < 1$.

Consider the real sequence $\sequence {b_n}$ defined by $\sequence {b_n} = \sequence {\size {a_n} }$.

Here, $\size {a_n}$ denotes either the absolute value of $a_n$, or the complex modulus of $a_n$.

Then:

$\displaystyle l = \limsup_{n \mathop \to \infty} {b_n}^{1/n}$

It follows from Terms of Bounded Sequence Within Bounds that for sufficiently large $n$,:

$b_n < \paren {l + \epsilon}^n$

By Sum of Infinite Geometric Sequence, the series $\displaystyle \sum_{n \mathop = 1}^\infty \paren {l + \epsilon}^n$ converges.

By the comparison test, $\displaystyle \sum_{n \mathop = 1}^\infty b_n$ converges.

Hence $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ converges absolutely by the definition of absolute convergence.

$\Box$


Divergence

Let $l > 1$.

Then we choose $\epsilon > 0$ such that $l - \epsilon > 1$.

Aiming for a contradiction, suppose that there exist an upper bound for the set:

$S := \set {n \in \N: \size {a_n}^{1/n} > l - \epsilon}$

Then for all sufficiently large $n$:

$\size {a_n}^{1/n} \le l - \epsilon$

However, this implies that:

$\displaystyle \limsup_{n \mathop \to \infty} \size {a_n}^{1/n} \le l - \epsilon$

which is false by the definition of $l$.

The set $S$, then, is not bounded.

This means that there exist arbitrarily large $n$ such that:

$\size {a_n} > \paren {l - \epsilon}^n$

Thus:

$\displaystyle \lim_{n \mathop \to \infty} \size {a_n} \ne 0$

and so $\displaystyle \lim_{n \mathop \to \infty} a_n \ne 0$.

Hence from Terms in Convergent Series Converge to Zero, $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ must be divergent.

$\blacksquare$


Notes

If $l = 1$, it is impossible to say, without further analysis, whether $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ converges absolutely, converges conditionally, or diverges.

If $\displaystyle \limsup_{n \mathop \to \infty} \size {a_n}^{1/n} = \infty$, then of course $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ diverges.


Sources