∵f（x）在点x=0的某一邻域内具有二阶连续导数，即f（x），f'（x），f''（x）在x=0的某一邻域均连续
且：
| lim |
| x→0 |
| f(x) |
| x |
∴f（x）=f（0）=0
| lim |
| x→0 |
| f(x)−f(0) |
| x |
∴f’（0）=0
∴
| lim |
| x→0 |
| f(x) |
| x2 |
| lim |
| x→0 |
| f’(x) |
| 2x |
| lim |
| x→0 |
| f’(x)−f’(0) |
| 2x |
| 1 |
| 2 |
∴
| lim |
| n→∞ |
f(
| ||
(
|
∴由比值判别法可知原级数绝对收敛
| lim |
| x→0 |
| f(x) |
| x |
| ∞ |
 |
| n＝1 |
| 1 |
| n |
| lim |
| x→0 |
| f(x) |
| x |
| lim |
| x→0 |
| f(x)−f(0) |
| x |
| lim |
| x→0 |
| f(x) |
| x2 |
| lim |
| x→0 |
| f’(x) |
| 2x |
| lim |
| x→0 |
| f’(x)−f’(0) |
| 2x |
| 1 |
| 2 |
| lim |
| n→∞ |
f(
| ||
(
|