| 1 | 1 | 1 | n | ||||
+ | +...+ | = | |||||
| √2*(√2+1) | (√2+1)*(√2+2) | (√2+n−1)*(√2+n) | √2*(√2+n) |
| 1 | 1 | ||
= | czyli OK | ||
| √2(√2+1) | √2(√2+1) |
| 1 | 1 | ||
+ | + ... + | ||
| √2(√2+1) | (√2+1)(√2+2) |
| 1 | n | ||
= | |||
| (√2 +n −1)(√2+n) | √2(√2+n) |
| 1 | 1 | ||
+ | + ... + | ||
| √2(√2+1) | (√2+1)(√2+2) |
| 1 | 1 | n+1 | |||
+ | = | ||||
| (√2 +n −1)(√2+n) | (√2 +n)(√2+n + 1) | √2(√2+n+1) |
| 1 | 1 | |||
L = | + | + ... + | ||
| √2(√2+1) | (√2+1)(√2+2) |
| 1 | 1 | ||
+ | teraz korzystamy z założenia czyli, | ||
| (√2 +n −1)(√2+n) | (√2 +n)(√2+n + 1) |
| n | 1 | |||
= | + | sprowadzamy do wspolnego mianownika | ||
| √2(√2+n) | (√2 +n)(√2+n + 1) |
| n(√2 + n + 1) + √2 | ||
= | = | |
| √2(√2 + n)(√2 + n + 1) |
| n2 + √2n + n + √2) + √2 | ||
= | = | |
| √2(√2 + n)(√2 + n + 1) |
| (n+1)(n+√2 | n+1 | |||
= | = | = P | ||
| √2(√2 + n)(√2 + n + 1) | √2)(√2+n+1) |