Proof
The proof of the convergence of a series Σan is an application of the comparison test. If for all n ≥ N (N some fixed natural number) we have then Since the geometric series converges so does by the comparison test. Absolute convergence in the case of nonpositive an can be proven in exactly the same way using
If for infinitely many n, then an fails to converge to 0, hence the series is divergent.
Proof of corollary: For a power series Σan = Σcn(z − p)n, we see by the above that the series converges if there exists an N such that for all n ≥ N we have
equivalent to
for all n ≥ N, which implies that in order for the series to converge we must have for all sufficiently large n. This is equivalent to saying
so Now the only other place where convergence is possible is when
(since points > 1 will diverge) and this will not change the radius of convergence since these are just the points lying on the boundary of the interval or disc, so
Read more about this topic: Root Test
Famous quotes containing the word proof:
“War is a beastly business, it is true, but one proof we are human is our ability to learn, even from it, how better to exist.”
—M.F.K. Fisher (1908–1992)
“It comes to pass oft that a terrible oath, with a swaggering accent sharply twanged off, gives manhood more approbation than ever proof itself would have earned him.”
—William Shakespeare (1564–1616)
“Ah! I have penetrated to those meadows on the morning of many a first spring day, jumping from hummock to hummock, from willow root to willow root, when the wild river valley and the woods were bathed in so pure and bright a light as would have waked the dead, if they had been slumbering in their graves, as some suppose. There needs no stronger proof of immortality. All things must live in such a light. O Death, where was thy sting? O Grave, where was thy victory, then?”
—Henry David Thoreau (1817–1862)