A company needs $\$ 3,900,000$ in 5 years in order to expand their factory. How much should the company invest each week if the investment earns a rate of $5.6 \%$ compounded weekly?
\[
\$
\]

Step 1 ：Given that the future value (FV) is $3,900,000, the annual interest rate (r) is 5.6% or 0.056 in decimal form, the number of compounding periods per year (n) is 52 (since the investment is compounded weekly), and the number of years (t) is 5. We want to find the payment per period (P).

Step 2 ：We use the formula for the future value of an annuity, which is: \(FV = P \times \frac{(1 + r/n)^{nt} - 1}{r/n}\)

Step 3 ：Rearranging the formula to solve for P gives us: \(P = FV \times \frac{r/n}{(1 + r/n)^{nt} - 1}\)

Step 4 ：Substituting the given values into the formula gives us: \(P = $3,900,000 \times \frac{0.056/52}{(1 + 0.056/52)^{52 \times 5} - 1}\)

Step 5 ：This simplifies to: \(P = $3,900,000 \times \frac{0.001076923076923077}{(1.001076923076923)^{260} - 1}\)

Step 6 ：Further simplifying gives: \(P = $4,198.461538461538 / (2.938618243487073 - 1)\)

Step 7 ：Finally, we calculate the value of P: \(P = $4,198.461538461538 / 1.938618243487073\)

Step 8 ：So, the company should invest approximately \(\boxed{$2,165.32}\) each week to reach their goal of $3,900,000 in 5 years.