Convergence - Ratio Test

Ratio test which is also known as D'Alembert's ratio is used to test for positive terms.

Let \(u_{1}+u_{2}+u_{3}+u_{4}+u_{5}+\cdot\cdot\cdot+u_{n}\) be a series of positive terms. Find expressions for \(u_{n}\) and \(u_{n+1}\), i.e. the \(n\)th and the \(n+1\)th term, and form the ratio \(\displaystyle\frac{u_{n+1}}{u_n}\). Determine the limiting value of this ratio as \(n \rightarrow\infty\).

If \(\displaystyle\lim_{n \rightarrow\infty} \frac{u_{n+1}}{u_n}<1\), the series converges

If \(\displaystyle\lim_{n \rightarrow\infty} \frac{u_{n+1}}{u_n}>1\), the series diverges

If \(\displaystyle\lim_{n \rightarrow\infty} \frac{u_{n+1}}{u_n}=1\), the series may converge or diverge and the test gives us no definite information

Example Question 1

Test the series \(\displaystyle\frac{1}{1}+\displaystyle\frac{3}{2}+\displaystyle\frac{5}{2^2}+\displaystyle\frac{7}{2^3}+\displaystyle\cdot\cdot\cdot\)

We first all decide on the pattern of the terms and hence write down the \(n\)th term. In this case \({u_{n}} = \displaystyle\frac{2n-1}{2^{n-1}}\). The \(n+1\)th term will then be the same with \(n\) replaced by \(n+1\)

i.e. \({u_{n+1}} = \displaystyle\frac{2n+1}{2^n}\)

therefore, \(\displaystyle\frac{u_{n+1}}{u_n} = \displaystyle\frac{2n+1}{2^n} \times \displaystyle\frac{2^{n-1}}{2n-1} = \displaystyle\frac{1}{2} \times \displaystyle\frac{2n+1}{2n-1} \)

We now have to find the limiting value of the ratio as \(n \rightarrow\infty\).

So, \(\displaystyle\lim_{n \rightarrow\infty} \displaystyle\frac{u_{n+1}}{u_n} = \displaystyle\lim_{n\rightarrow\infty} \displaystyle\frac{1}{2} \times \displaystyle\frac{2n+1}{2n-1} = \displaystyle\lim_{n \rightarrow\infty} \displaystyle\frac{1}{2} \times \displaystyle\frac{2+1/n}{2-1/n} = \displaystyle\frac{1}{2} \times \displaystyle\frac{2+0}{2-0}= \frac{1}{2} \)

Because in this case, \(\displaystyle\lim_{n \rightarrow\infty} \displaystyle\frac{u_{n+1}}{u_n}<1\), we know that the given series is \(convergent\).