Special Case in Ratio Test for Series

Question

We all know the Ratio Test in series which is one of many tests that are used to determine whether a Series is Converges or Not (Most Of The Time Work). Here is a link that describe Ratio test "In Short". Ratio Test

I have example : Suppose we have $U_n$=$2^{-n+({-1})^{n+1}}$ and we want to know whether the series $$\sum_{n=1}^\infty U_n$$ Is converges Or Not. $\\$

If we Use the Ratio Test then $\frac{U_{n+1}}{U_n}$=$2^{2({-1})^{n}-1}$, and we can Rewrite it as$$ \frac{U_{n+1}}{U_n} = \begin{cases} 2, & \text{if $n$ is even} \\ \frac{1}{8}, & \text{if $n$ is odd} \end{cases}$$ And I think if the Proudct of the two number $2$ and $\frac{1}{8}$ is exactly less than $1$ then the series is Converges . in general Form if the fraction $\frac{U_{n+1}}{U_n}$ Can be written of the Form of $$ \frac{U_{n+1}}{U_n} = \begin{cases} A_n, & \text{if $n$ is even} \\ B_n, & \text{if $n$ is odd} \end{cases}$$ and we have $$\lim_{n\to \infty}{A_n}= a \text{ and} \lim_{n\to \infty}{B_n}= b $$ then if $ 0 < a.b < 1 $then the series $\sum U_n$ Is Converges. What do you think in this idea ?

for Now it is just true when $U_n >0$, I did not study it when $U_n <0$

So Give me your Opinion whether my idea is True Or False

I know that we can use another Test to prove the convergence of the first series .. but i want it in this way .. with thanks — Yasser Milato, Dec 24 '17 at 16:09
https://math.stackexchange.com/questions/1164138/ratio-test-with-limsup-vs-lim — A. Thomas Yerger, Dec 24 '17 at 16:15
Indeed, it is correct. And if you showed it with the odd/even subsequences, then your proof is likely correct. — gniourf_gniourf, Dec 24 '17 at 16:16

score 1 · Accepted Answer · answered Dec 24 '17 at 17:09

Note that

$$\sum_{n=1}^{2N} U_n=\sum_{n=1}^N U_{2n}+\sum_{n=1}^N U_{2n-1}\tag1$$

We assume that $U_n\ge 0$,

If the limits $\lim_{n\to \infty}\frac{U_{2n+2}}{U_{2n+1}}$ and $\lim_{n\to \infty}\frac{U_{2n+1}}{U_{2n}}$ exist and are finite, then the ratio test guarantees that the first sequence of partial sums on the right-hand side of $(1)$ converges when

$$\begin{align} \lim_{n\to\infty}\frac{U_{2(n+1)}}{U_{2n}}&=\lim_{n\to \infty}\left(\frac{U_{2n+2}}{U_{2n+1}}\frac{U_{2n+1}}{U_{2n}}\right)\\\\ &=\left(\lim_{n\to \infty}\frac{U_{2n+2}}{U_{2n+1}}\right)\,\left(\lim_{n\to \infty}\frac{U_{2n+1}}{U_{2n}}\right) \\\\&<1\tag 2 \end{align}$$

Similary, if the limits $\lim_{n\to \infty}\frac{U_{2n}}{U_{2n-1}}$ and $\lim_{n\to \infty}\frac{U_{2n+1}}{U_{2n}}$ exist and are finite, then the ratio test guarantees that the second sequence of partial sums on the right-hand side of $(1)$ converges when

$$\begin{align} \lim_{n\to\infty}\frac{U_{2(n+1)-1}}{U_{2n-1}}&=\lim_{n\to \infty}\left(\frac{U_{2n+1}}{U_{2n}}\frac{U_{2n}}{U_{2n-1}}\right)\\\\ &=\left(\lim_{n\to \infty}\frac{U_{2n+1}}{U_{2n}}\right)\,\left(\lim_{n\to \infty}\frac{U_{2n}}{U_{2n-1}}\right) \\\\&<1\tag3 \end{align}$$

Hence, if $(2)$ and $(3)$ hold, then the sequence of partial sums on the left-hand side of $(1)$ converges.

We conclude that if the product of the limits $\lim_{n\to\infty}\frac{U_{n+1}}{U_n}$ for $n$ even and $\lim_{n\to\infty}\frac{U_{n+1}}{U_n}$ for $n$ odd is less than $1$, then the series $\sum_{n=1}^\infty U_n$ converges.

Special Case in Ratio Test for Series

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