It looks like you've uploaded a mathematical problem involving sums. The problem contains multiple summations to evaluate. Let's break them down one by one:

1. i) Evaluate the sum: $\sum_{k=-3}^{n} (2^k - 2^{k-1})$

2. ii) Evaluate the sum: $\sum_{k=1}^{10^6} \left( \frac{1}{k} - \frac{1}{k+1} \right)$

3. iii) Evaluate the sum:

   \quad \text{with} \quad j \neq 0, 1$$

4. iv) Evaluate the sum: $\sum_{i=1}^{n} [F(i) - F(i-1)]$

5. v) Evaluate the sum: $\sum_{j=1}^{n} (4j + 3) \cdot \frac{1}{n^2}$

6. vi) Evaluate the sum: $\sum_{i=1}^{n} \frac{4i^2 (i-1)}{n^4}$

7. vii) Evaluate the sum: $\sum_{i=1}^{10^2} \frac{2}{i(i-2)}$

8. viii) Evaluate the sum: $\sum_{m=0}^{2019} (1 + m)$

9. ix) Evaluate the sum: $\sum_{k=1}^{n} \left[ (3^k - 3^{k-1})^2 - (3^{k-1} - 3^{k+1})^2 \right]$

Would you like me to walk you through the solutions step-by-step, or would you prefer specific help with any particular sum? Let me know!