# Sum disappearing when we assume constant some elements to be constant over time [closed]

I have the dividend discount model, which is the following expression:

$$P_{j,t} = \sum_{\tau=1}^{\infty}D_\tau(1+g)^\tau(1+r)^{-\tau}=\frac{D_{\tau+1}}{r-g}$$

Where $$D_t$$, is the dividend at time $$t$$ ,$$g$$ represents the constant growth over time and $$r$$ represents the required rate of return which is assumed to be constant over time too.

My questions are: why under constant growth and constant required rate of return can we re-write it this way. Why does the sum sign disappears?

EDIT 1:

@ZRH's solution is correct. I also found a website with more intermediate steps here: http://www.calculatinginvestor.com/2011/05/18/gordon-growth-model/

Thank you

• It is a property of Geometric Series that allows the summation to be written in closed form this way en.wikipedia.org/wiki/Geometric_series#Formula – Alex C Mar 23 '19 at 15:25
• Very important to note that this equation only holds if $r>g$. – Alex C Mar 23 '19 at 15:47
• @AlexC Thank you. The assumption that $r>g$ is a general assumption of the model, otherwise, the company would grow bigger than the entire economy in perpetuity – Adrian Mar 23 '19 at 16:19
• Yes, it can be interpreted as an economic assumption. But it is also mathematically necessary for the infinite sum to converge. – Alex C Mar 23 '19 at 17:21
• Ah. I see. That makes sense. Thank you for the additional explanations. Been a great help – Adrian Mar 23 '19 at 17:23

Using Alex C's link, and further assuming the dividends $$D_\tau$$ to be constant (else you cannot really come up with a simple formula) you get:
$$P=D\sum_{\tau=1}^\infty \left(\frac{1+g}{1+r}\right)^{\tau}=\frac{\frac{1+g}{1+r}}{1-\frac{1+g}{1+r}}=D\frac{1+g}{r-g}$$
I presume that in your above formula, you mean $$D(1+g)$$ when you write $$D_{\tau+1}$$, as after executing the summation there should not be an index $$\tau$$ anymore