The root test has to do with when a series of complex numbers converges. I am assuming the reader has been exposed to infinite series. However, this that I am about to explain is a little more general than what is usually seen in calculus.
Theorem 2.4.1 Let a_{k} ∈ F^{p}and consider ∑ _{k=1}^{∞}a_{k}. Then this series converges absolutely if

The series diverges spectacularly if limsup_{k→∞}
Proof: Suppose first that limsup_{k→∞}

Hence there exists N such that if k ≥ N, then

and so, by the Weierstrass M test applied to the series of constants, the series converges and also converges absolutely. If

then letting r > R > 1, it follows that for infinitely many k,

and so there is a subsequence which is unbounded. In particular, the series cannot converge and in fact diverges spectacularly. In case that the limsup = 1, you can consider ∑ _{n=1}^{∞}
This is a major theorem because the limsup always exists. As an important application, here is a corollary.
Corollary 2.4.2 If ∑ _{k}a_{k} converges, then limsup_{k→∞}
If the sequence has values in X a complete normed linear space, there is no change in the conclusion or proof of the above theorem. You just replace