- An infinite series is convergent if from and after some fixed term, the ratio of each term to the preceding term is numerically less than some quantity which is itself numerically less than unity.
Let the series beginning from the fixed term be denoted by
u1+u2+u3+u4+⋯,
and let
$$\frac{u_2}{u_1}where r<1.
Then
\begin{align*}
&u_1+u_2+u_3+u_4+\dotsb\\
&=u_1\left(1+\frac{u_2}{u_1}+\frac{u_3}{u_2}\cdot\frac{u_2}{u_1}+\frac{u_4}{u_3}\cdot\frac{u_3}{u_2}\cdot\frac{u_2}{u_1}+\dotsb\right)\\
&\end{align*}
that is, <u11−r, since r<1.
Hence the given series is convergent.
Does that mean that the series 1+12+13+⋯ should be a convergent one?
Sn=11+12+13+14+⋯
Here, we have
unun−1=1/n1/(n−1)=n−1n=1−1n∴uun−1<1∴Series should be convergent
Answer
You need unun−1<r<1 for some r<1 and all n.
For any r<1 we can find r<unun−1<1. So we can not find an appropriate r<1.
So we failed the hypothesis. The ratio test fails.
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