What is the interval of convergence when radius of convergence is 0?

converges if and only if x = 0. Therefore the radius of convergence is 0 and the interval of convergence is [0, 0].

Can the radius of convergence be undefined?

Even if we try to define f(0) , then f'(0) will be undefined. So either the Maclaurin series is undefined or it will only describe f(0) and have a zero radius of convergence.

Can radius of convergence be less than 1?

(Sometimes we say the sum converges to infinity, but usually we say it diverges). For X smaller than one and bigger than minus one, the sum can be done. The radius of convergence for this function is one.

Do all power series converge at 0?

First, we prove that every power series has a radius of convergence. be a power series. There is an 0 ≤ R ≤ ∞ such that the series converges absolutely for 0 ≤ |x − c| < R and diverges for |x − c| > R.

Is it possible for a power series to have an empty interval of convergence?

The interval of convergence is never empty Every power series converges for some value of x. That is, the interval of convergence for a power series is never the empty set.

What happens if radius of convergence is infinity?

The number R given in Theorem 73 is the radius of convergence of a given series. When a series converges for only x=c, we say the radius of convergence is 0, i.e., R=0. When a series converges for all x, we say the series has an infinite radius of convergence, i.e., R=∞.

What happens when radius of convergence is infinity?

On the boundary, that is, where |z − a| = r, the behavior of the power series may be complicated, and the series may converge for some values of z and diverge for others. The radius of convergence is infinite if the series converges for all complex numbers z.

What happens if the Ratio Test equals 0?

r = 0 implies the power series is convergent for all x values, and r = ∞ implies the power series is divergent always. Again we have the case that r = 0 < 1, hence we can conclude that the power series converge for all x values.

Do all convergent series converge to 0?

Yes, one of the first things you learn about infinite series is that if the terms of the series are not approaching 0, then the series cannot possibly be converging. This is true.

Does a series converge if the limit is 0?

If the limit is zero, then the bottom terms are growing more quickly than the top terms. Thus, if the bottom series converges, the top series, which is growing more slowly, must also converge. If the limit is infinite, then the bottom series is growing more slowly, so if it diverges, the other series must also diverge.

Is there a power series with radius of convergence of 0?

The ratio test tells us that the series diverges: n! x n ( n − 1)! x n − 1 = n x. As long as | x | > 0, there is some N such that this is > 1 for all n > N, so the series diverges. Yes, the radius of convergence can be 0, e.g., ∑ n! x n. It will, of course, still converge at x = 0, but nowhere else.

How does convergence depend on values of X?

In the discussion of power series convergence is still a major question that we’ll be dealing with. The difference is that the convergence of the series will now depend upon the values of x that we put into the series. A power series may converge for some values of x

Is the power series always converge for x x?

It is important to note that no matter what else is happening in the power series we are guaranteed to get convergence for x = a x = a. The series may not converge for any other value of x x, but it will always converge for x = a x = a.

Which is an example of convergence of a power series?

Before getting into some examples let’s take a quick look at the convergence of a power series for the case of x = a x = a. In this case the power series becomes, and so the power series converges. Note that we had to strip out the first term since it was the only non-zero term in the series.