| (−1)n+1*n | ||
∑ | n=1 do n=∞ | |
| 3n |
| (−1)n+1*n | 1 | (−1)n+1n | ||||
S = ∑1 | = | + ∑2 | = | |||
| 3n | 3 | 3n |
| 1 | (−1)n+2(n+1) | |||
= | + ∑1 | = | ||
| 3 | 3n+1 |
| 1 | (−1)n+1n + (−1)n+1 | |||
= | − ∑1 | = | ||
| 3 | 3n+1 |
| 1 | (−1)n+1n | (−1)n+1 | ||||
= | − ∑1 | − ∑1 | = (*) | |||
| 3 | 3n+1 | 3n+1 |
| −1 | ||
Zajmijmy się drugim szeregiem, jest to szereg geometryczny o ilorazie q = | ||
| 3 |
| a1 | ||
S = ∑qn= | ||
| 1 − q |
| (−1)n+1 | −1 |
| |||||||||||||
∑1 | = ∑1( | )n+1 = | = | ||||||||||||
| 3n+1 | 3 |
|
| 1 | 3 | 1 | ||||
= | * | = | ||||
| 9 | 4 | 12 |
| 4 | 1 | (−1)n+1n | 1 | |||||
(*) = | − | ∑1 | − | = | ||||
| 12 | 3 | 3n | 12 |
| 1 | 1 | |||
= | − | S | ||
| 4 | 3 |
| 1 | 1 | |||
S = | − | S | ||
| 4 | 3 |
| 4 | 1 | ||
S = | |||
| 3 | 4 |
| 3 | ||
S = | ||
| 16 |