Power sum MASTER CLASS: How to sum quadrillions of powers ... by hand! (Euler-Maclaurin formula)

What people said

Strong rec K.defaoite · 2020-07 · Mathematics Stack Exchange · on “Sum of Squares $i^2$”

Not super rigorous, but this amazing animation [this video] is a nice visual proof. In fact watch the entire video, it's fantastic.

Strong rec K.defaoite · 2021-04 · Mathematics Stack Exchange · on “Induction on $S(n,k)$”

This is a well known problem in mathematics. See Faulhaber's formula [LINK] and this excellent Mathologer video [this video] .

Recommends Toby Mak · 2019-11 · Mathematics Stack Exchange · on “Can you go from the integral of x squared to the summation of x squared?”

Mathologer [this video] did a whole episode on your topic. Although the derivation does not use calculus, it is still quite useful to what you are asking. Your part starts at 9:28.

Recommends Jonah · 2020-03 · Mathematics Stack Exchange · on “Geometric interpretation for sum of fourth powers”

This mathologer video [this video] is not exactly an answer to your question, but goes into the general formula for any power in an accessible, visual way.

Recommends K.defaoite · 2020-05 · Mathematics Stack Exchange · on “Simple proof by induction problems”

Also, I'm not sure induction is the best method for your other problem. Learning about integer power sums, which Mathologer has a great video [this video] on would help you with this problem.

Recommends Antoni Parellada · 2023-05 · Mathematics Stack Exchange · on “What is the simple dependence of the diagonals (or columns) of the Faulhaber matrix on the first entry (Bernoulli numbers)?”

In this presentation by Mathologer [this video] you can find the following slide: [LINK] relating the sum of the $n$ first $k$ powers of integers, i.e. $S_k$ and the Faulhaber matrix diagonals (which correspond to the columns highlighted on the image above. Here is the corresponding Faulhaber matrix: S 0 1 0 0 0 0 0 0 S 1 1/2 1/2 0 0 0 0 0 S 2 1/6 1/2 1/3 0 0 0 0 S 3 0 1/4 1/2 1/4 0 0 0 S 4 -1/30 0 1/3 1/2 1/5 0 0 S 5 0 -1/12 0 5/12 1/2 1/6 0 S 6 1/42 0 -1/6 0 1/2 1/2 1/7 The patterns along the first four columns are well explained: The harmonic sequence, $1/2$ 's, $k/12$ , $0$ 's, and then the following statement: Jacob Bernoulli discovered something amazing: The numbers in the different columns

Recommends feli_x · 2023-08 · Mathematics Stack Exchange · on “Search for a geometrical way to calculate the summations $\sum_{k=1}^n k^p$ ($p=1,2,3...$)”

There is an analogous geometrical view for the case $p=1$ in 2 dimensions. Look at Fig. 3 [LINK] , which illustrates the view for $n=4$ . The pyramid in 3d corresponds to the triangle in 2d. The sum $1+2+3+4$ can be calculated as the area of the pseudo-triangle. One can approximate this area as the area of the triangle which has the dashed line as one side and afterwards add 4 half squares extending beyond the dashed line. $S_1(4) = \frac{1}{2} \cdot 4 \cdot 4 + 4 \cdot \frac{1}{2} = 8 + 2 = 10$ . For arbitrary $n$ one arrives at the "Gauß-formula" $S_1(n) = \frac{1}{2} n \cdot n + \frac{n}{2} = \frac{1}{2} n \cdot (n+1)$ . This geometric proof is also shown in the video Power sum master cla

Recommends feli_x · 2023-08 · Mathematics Stack Exchange · on “Search for a geometrical way to calculate the summations $\sum_{k=1}^n k^p$ ($p=1,2,3...$)”

There exist geometrical proofs for the calculation of the sums $S_p(n):=\sum_{k=1}^nk^p$ for $p=1$ and $p=2$ (e.g. compilation of proofs for sum of squares and cubes [LINK] , understanding some PWoWs [LINK] ). The core question of this thread is: Does there exist a geometry-based method to prove the formulas of the above given summations, which is applicable for all integers $n$ and $k$ ? I found geometrical proofs for $p=2$ and $p=3$ , which I find beautiful. (I am sure other persons found these visualizations before, there are hints on this e.g. in the video Power sum master class [this video] .) The method of this proof for $p=2$ consists in interpreting the sum $1^2+2^2+3^3+...$ as the volume

Cites Toby Mak · 2019-11 · Mathematics Stack Exchange · on “When is it appropriate to find the integral of a function?”

The problem with using integration in your first example is that the function is discrete . For the whole of day $1$ , the population of ants is $1$ ; for the whole of day $2$ , the population of ants is $1+4=5$ , and so on. What you would have to integrate to get the correct result is a discrete function $f(x)$ , such that $f(x) = 0$ for $-1 ≤ x < 0$ , $f(x) = 1$ for $0 ≤ x < 1$ , $f(x) = 4$ for $1 ≤ x < 2$ and so on. For your second example, the case is the same. Because the total amount of sunlight is only given for each day $d$ , and not for all the times in the year, then using integration will not give the correct result. Since the number of summations is large ( $365$ days), this will