Convergence of Ratio Test for Complex Numbers z in C

In summary: Thank you for pointing it out. In summary, the ratio test involves the ratio of coefficients, not terms, and in this case, the ratio is $|\frac{\pi z}{n+2}|$.
  • #1
Dustinsfl
2,281
5
$$\sum\limits_{n = 0}^{\infty}\frac{2\pi^{n+1}M}{(n+1)!}z^n$$

So we get $\lim\limits_{n\to\infty}\left|\frac{\pi z^n}{n+2}\right|$.

This converges but I don't see how. z is in C.
 
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  • #2
Ratio test involves $\left|\frac{a_{n+1}}{a_n}\right|$ where $a_n$, $a_{n+1}$ are coefficients of the series. In other words, $z$ should not be in the ratio.
 
  • #3
Evgeny.Makarov said:
Ratio test involves $\left|\frac{a_{n+1}}{a_n}\right|$ where $a_n$, $a_{n+1}$ are coefficients of the series. In other words, $z$ should not be in the ratio.

Even though z has a power of n?

---------- Post added at 08:07 PM ---------- Previous post was at 07:21 PM ----------

dwsmith said:
Even though z has a power of n?

I see my mistake. It should be z not z^n
 
  • #4
dwsmith said:
It should be z not z^n
In a power series $\sum a_nz^n$, the numbers $a_n$ are called coefficients and the products $a_nz^n$ are called terms. The ratio test involves the ratio of coefficients, not terms. The ratio $a_{n+1}/a_n$ has nothing to do with $z$.
 
  • #5
The ratio test of a series $\displaystyle \sum_{n=0}^{\infty} a_{n}$ is a test on the quantity $\displaystyle r_{n}=|\frac{a_{n+1}}{a_{n}}|$. In this case is $\displaystyle a_{n}= \frac{2\ \pi^{n+1}\ M}{(n+1)!}\ z^{n}$ so that $\displaystyle r_{n}= |\frac{\pi\ z}{n+2}|$...

Kind regards

$\chi$ $\sigma$
 
  • #6
OK, I had a momentary lapse of reason. Of course, ratio test is a test for any series, not just power series, so the concept of coefficient does not apply here. You are right, chisigma and dwsmith.
 

FAQ: Convergence of Ratio Test for Complex Numbers z in C

What is the Ratio Test?

The Ratio Test is a mathematical test used to determine the convergence or divergence of an infinite series. It compares the terms of the series to the terms of a geometric series to determine if the series is convergent or divergent.

How do you perform the Ratio Test?

To perform the Ratio Test, you take the limit of the absolute value of the ratio of the (n+1)th term to the nth term of the series. If this limit is less than 1, then the series is convergent. If the limit is greater than 1, the series is divergent. If the limit is equal to 1 or the limit does not exist, the test is inconclusive.

What is a geometric series?

A geometric series is a series in which the ratio of each term to the previous term is constant. It has the form a + ar + ar^2 + ar^3 + ... where a is the first term and r is the common ratio.

When should you use the Ratio Test?

The Ratio Test should be used when the terms of the series involve powers of n or factorials. It is also useful for determining the convergence or divergence of alternating series.

What is the difference between the Ratio Test and the Root Test?

Both the Ratio Test and the Root Test are used to determine the convergence or divergence of infinite series. The main difference is that the Ratio Test compares the terms of the series to a geometric series, while the Root Test compares the terms to a p-series. In some cases, one test may be easier to use than the other, so it is important to know both tests.

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