How to determine sum of an alternating power series and to prove that sum is positive

Question

I am working on a problem involving an alternating power series as follows:

$$\sum_{i=0}^{a-2} (-1)^{a+b-i-2}(a+b-i-1)x^{a+b-i-2}$$

$a$ and $b$ is constant with $0<x<1$

I would like to determine an expression which is used to calculate the sum of this series and to prove the expression is positive. However, I have no idea how to start. Does anyone have any hints or ideas about this problem? Please let me know. I really appreciate your help.

Answer 1

Hint: Some transformations to make the expression easier manageable.

We obtain \begin{align*} \sum_{i=0}^{a-2}&(-1)^{a+b-i-2}(a+b-i-1)x^{a+b-i-2}\\ &=\sum_{i=0}^{a-2}(b+1+i)(-x)^{b+i}\tag{1}\\ &=\sum_{i=b+1}^{a+b-1}i(-x)^{i-1}\tag{2}\\ &=-D_x\sum_{i=b+1}^{a+b-1}(-x)^i\tag{3}\\ &=D_x\left(\frac{(-x)^{a+b}-(-x)^{b+1}}{1+x}\right)\tag{4} \end{align*}

Comment:

• In (1) we revert the summation order by transforming the index $i \rightarrow a-2-i$

• In (2) we shift the index $i \rightarrow i+b+1$

• In (3) we apply the differential operator $D_x:=\frac{d}{dx}$

• In (4) we use the generalised formula for the finite geometric series