Question

State and prove cauchy's nth root test

Answer 1

The root test uses the number where"lim sup"denotes the limit superior,possibly ∞.Note that if converges then it equals C& may be used in the root test instead.The root test states that:
If C<1 then the series converges absolutely,
if C>1 then the series diverges
if C=1 & limit approaches strictly from above then the series diverges.
otherwise the test is inconclusive (the series may diverge,converge absolutely or converge conditionally)
Proof;
the proof of the convergence of a series∑an is an application of the comparison test.If for all n≥(N some fixed natural number)we have then.Since the geometric series converges so does by the comparison test.Hence ∑an converges absolutely.Note that...implies that...for almost all
If for infinitely many n,then an fails to converge to 0,hence the series is divergent.

Answer 2

Answer:

State and prove Cauchy's nth root test are shown below:

Step-by-step explanation:

Theorem

The root test utilizes the number where "lim sup" represents the limit superior, possibly ∞.

Note that if converges then it equals C and may be used in the root test instead.

The root test states that:

If$C < 1$ then the series converges absolutely,

if $C > 1$then the series diverges

if $C=1$ and limit approaches strictly from above then the series diverges.

otherwise, the test is inconclusive (the series may diverge, converge absolutely or converge conditionally).

Proof:

The proof of the convergence of a series ∑$a_{n}$ is an application of the comparison test. If for all n≥(N some fixed natural number)we have then. Since the geometric series converges so does the comparison test. Hence ∑$a_{n}$ converges absolutely.

If for infinitely many n, then $a_{n}$ fails to converge to 0, hence the series is divergent.

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