1. ## ratio test

I was told to show that if 0<r<1 and p is any positive integer, sum of n = p+1 to infinity n(n-1)...(n-p)r^n is convergent using ratio test.

when I get Vn+1/Vn I got (n+1)/(n-p)r^n which tells me the series is divergent as it goes to infinity??? can you help me out???

and also can you help to figure this out using the root test??

Thanks

2. I would use the ratio test on the sum of n = 1 to infinity, the algebra is easier. Clearly this should be greater than the sum of n = p+1 to infinity, so if it converges for the larger series, the smaller series must converge too.

3. We find $\displaystyle \frac{v_{n+1}}{v_n} = \frac{n+1}{n-p}r$ which converges to $\displaystyle r$. The root test gives that the series is convergent.

4. Originally Posted by mathsohard
I was told to show that if 0<r<1 and p is any positive integer, sum of n = p+1 to infinity n(n-1)...(n-p)r^n is convergent using ratio test.

when I get Vn+1/Vn I got (n+1)/(n-p)r^n which tells me the series is divergent as it goes to infinity??? can you help me out???

and also can you help to figure this out using the root test??

Thanks
If you put $\displaystyle a_n:=n(n-1)\cdot\ldots\cdot (n-p)r^n$ , then $\displaystyle \frac{a_{n+1}}{a_n}=\frac{n+1}{n-p}\,r\xrightarrow [n\to\infty] {} \, r<1$ and the series converges.

Tonio