Here is a somewhat standard presentation that might be taught in intermediate macroeconomics at US universities.
Let $p_t$ be the price of a unit of capital at the end of period $t$. Suppose that production takes place at the start of each period. After production takes place, depreciation occurs. Let us say that the depreciation rate is constant at $\delta$. Let us also say that the real interest rate across any two consecutive periods is $r$. Assume also that the price of each unit of output is equal to $1.
If so, we would expect that
$p_1=\frac{1}{1+r}[MP^k_2+(1-\delta)p_2]$.
This has a nice interpretation. The price of capital at the end of period 1 equals: The present discounted value of the additional output it can give us at the start of period 2, plus the after-depreciation resale value of that unit of capital at the end of period 2.
We can write down an analogous expression for $p_2$:
$p_2=\frac{1}{1+r}[MP^k_3+(1-\delta)p_3]$
Now substitute this latter expression for $p_2$ into our first equation above, to get:
$p_1=\frac{1}{1+r} \left\{ [MP^k_2+(1-\delta)\frac{1}{1+r}[MP^k_3+(1-\delta)p_3] \right\}=\frac{1}{1+r}[MP^k_2+\frac{1-\delta}{1+r}MP^k_3+\frac{(1-\delta)^2}{1+r}p_3]$
This last expression has again a nice interpretation. The price of capital at the end of period 1 equals: The present discounted value of the additional output it can give us at the start of period 2, plus the additional output it can give us at the start of period 3, plus the resale value of that unit of capital at the end of period 3.
If we keep going, plugging in expressions for $p_3$, $p_4$, ..., eventually we'll find that
$p_1=\frac{1}{1+r}[MP^k_2+\frac{1-\delta}{1+r}MP^k_3+(\frac{1-\delta}{1+r})^2MP^k_4+(\frac{1-\delta}{1+r})^3MP^k_5+\dots]$
This has the desired result: The price of a unit of capital today is simply equal to the present value of the future stream of income that this unit of capital will generate, appropriately taking into account depreciation.
This generalizes to a broader principle employed not just in standard macroeconomics but in the real world too: The price of any asset should be equal simply to the present value of the future stream of income that the asset will generate.