What happens if the ratio test equals 1?

What happens if the ratio test equals 1?

The ratio test states that: if L < 1 then the series converges absolutely; if L > 1 then the series is divergent; if L = 1 or the limit fails to exist, then the test is inconclusive, because there exist both convergent and divergent series that satisfy this case.

What does the alternating series test say?

The alternating series test can only tell you that an alternating series itself converges. The test says nothing about the positive-term series. In other words, the test cannot tell you whether a series is absolutely convergent or conditionally convergent.

Is the converse of the ratio test true?

The converse of this theorem is not true. Theorem (The Ratio Test) Let \begin{align*}\sum_{an}\end{align*} be a series of non-zero \begin{align*}\mathrm{numbers}^*\end{align*}. (A) If \begin{align*}\lim_{n \to \infty}\ \left |\frac{a_n+1}{a_n} \right |=\alpha < 1\end{align*}, then the series is absolutely convergent.

How do you know when to use the ratio test?

We will use the ratio test to check the convergence of the series. if L<1 the series converges absolutely, L>1 the series diverges, and if L=1 the series could either converge or diverge.

How many tests of convergence are there?

exists, there are three possibilities: if L > 1 the series converges. if L < 1 the series diverges. and if L = 1 the test is inconclusive.

Which among the following test is useful to examine the convergence of alternating series?

the Leibniz test
The alternating series test (also known as the Leibniz test), is type of series test used to determine the convergence of series that alternate. Keep in mind that the test does not tell whether the series diverges.

Is Root Test Stronger Than ratio test?

Strictly speaking, the root test is more powerful than the ratio test. In other words, any series to which we can conclusively apply the ratio test is also a series to which we can conclusively apply the root test, and in fact, the limit of the sequence of ratios is the same as the limit of the sequence of roots.