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 $convergent$.