1. The first problem asks to expand the sigma notation $$\sum_{i=1}^5 \frac{2^2}{i}$$ into a series. 2. The formula for sigma notation expansion is to substitute each integer value of $i$ from the lower limit to the upper limit into the expression and sum the results. 3. Here, $2^2 = 4$, so the series is: $$\frac{4}{1} + \frac{4}{2} + \frac{4}{3} + \frac{4}{4} + \frac{4}{5}$$ 4. Evaluating each term: - $\frac{4}{1} = 4$ - $\frac{4}{2} = 2$ - $\frac{4}{3} \approx 1.333$ - $\frac{4}{4} = 1$ - $\frac{4}{5} = 0.8$ 5. Adding these gives the sum: $$4 + 2 + 1.333 + 1 + 0.8 = 9.133$$ --- 6. The second problem asks to expand the sigma notation $$\sum_{i=2}^5 (i+4)^2$$ into a series. 7. We substitute $i=2,3,4,5$ into the expression $(i+4)^2$: - $(2+4)^2 = 6^2 = 36$ - $(3+4)^2 = 7^2 = 49$ - $(4+4)^2 = 8^2 = 64$ - $(5+4)^2 = 9^2 = 81$ 8. The series is: $$36 + 49 + 64 + 81$$ 9. Adding these gives the sum: $$36 + 49 + 64 + 81 = 230$$