I'm Working on an econ otimization problem (it's part of a model on david romer's advanced macro book. chapter 7, page 318). And my calc/algebra background is weak.
The optimization s as follows:
Min $\sum_{t=0}^\infty q_t(p_i - p_t^*)^2 $ , w.r.t $p_i$
What I would normally do is:
$$\sum_{t=0}^\infty 2q_t(p_i - p_t^*)1 =0 $$
$$\sum_{t=0}^\infty 2q_t p_i =\sum_{t=0}^\infty 2q_t(p_t^*) $$
$$2p_i\sum_{t=0}^\infty q_t =2\sum_{t=0}^\infty q_tp_t^* $$
$$p_i=\frac{\sum_{t=0}^\infty q_tp_t^*}{\sum_{t=0}^\infty q_t}$$
The correct step, on the book, is : $$p_i=\sum_{t=0}^\infty \frac{q_t}{\sum_{\tau=0}^\infty q_{\tau}} p_t^* $$
So:
Where does $ \sum_{\tau=0}^\infty q_{\tau} $ term comes from, and how does it end up inside the summation?
Isn't $ \sum_{\tau=0}^\infty q_{\tau} = \sum_{t=0}^\infty q_t$, as both go from zero to infinity ? What is the most intuitive form of understanding why this happens?
This could be a bit broad: I've only studied calculus 1 a few years ago. What kind of material covers these specific kinds of derivatives applied on summation and integrals ?
Thanks and sorry in advance for the math typing/english. First time using mathjax and stackexchange :)
Your answer is the same as the book but they are using two different summation indexes to avoid confusing them.
If the book had written it like this:
$p_i=\sum_{t=0}^\infty \frac{ q_t}{\sum_{t=0}^\infty q_t}p_t^*$
then this could be read as
$p_i= \frac{ q_0}{\sum_{t=0}^\infty q_0}p_0^* + \frac{ q_1}{\sum_{t=0}^\infty q_1}p_1^* + \sum_{t=2}^\infty \frac{ q_t}{\sum_{t=0}^\infty q_t}p_t^*$