Many people know the story of Gauss when he was a child. The story is about ten years old. The teacher had a difficult problem in arithmetic class: "What? 1 to? Write down the integer of 100 and add it up! The calculation method of Gaussian is:1+100 =10, 2+99 =10, 3+98 =10/kloc-. 50× 10 1=5050。
This is the summation of series, which we all learned in high school. It is expressed in mathematical form.
However, this method cannot be used with the sum of squares sequence, and it is difficult to make it by induction, so it is not universal.
Here is a general method that I found myself. With a simple integral formula, I can easily solve the formula of k power and sequence.
The relationship between the formula of sum of front and rear powers
K is an integer greater than or equal to 0, and the formula of power sum sequence of k has the following relationship with the formula of power sum sequence of (K+ 1).
1. Relationship between the formulas of power sum before and after.
C is a constant to be determined.
2. When n is 1, the sum is always 1.
3.？ No expression has a constant.
There is no constant in the expression of.
Knowing these three relationships, we can calculate the sum of series from 0 power to k power.
The sum of 1.0 power series
Because the power of 0 of each number is equal to 1, so
This conclusion can be easily drawn without using other algorithms.
Second, the sum of 1 power series
1 power sum sequence is a problem encountered by Gaussian, and the result is
Here, it's a little more complicated to use my method. I have to substitute the above formula to the power of 0 into this step.
When n= 1
therefore
Finally, the formula of the power sum sequence of 1 is
Cubic and quadratic series
The expression of the sum of squares sequence is as follows. Some people use mathematical induction to get this formula, but I think it is more difficult.
Repeat the above steps roughly according to my method, and you will get the answer soon.
When n= 1
therefore
So the sum of squares sequence formula is
Fourth and third power sum sequence
The formula of cubic power sum sequence is
We substitute the formula of sum of squares here, and the same result can be obtained in the same step.
When n= 1
therefore
The formula of cubic sum of squares series is
Summation of quintic and quartic power squares series
The formula of quartic power sum sequence is
? At present, I haven't found the proof method of this formula given by others, so it should be difficult for me to find out this formula.
We still use our method to calculate.
When n= 1
therefore
So the formula of quartic square and sequence is
As you can see, my method is simple and universal, and I can quickly calculate the K-power and sequence formula with simple integral formula.