Wolfram Language Coding

June 7, 2023

Anonymous

In the following every time I input a task/query/problem, I want you 1) write down the relevant mathematical formulas 2) develop and display well-documented Wolfram Language code that solves the problem and displays the code in a terminal-like way (with proper line length) 2) evaluate the code on the Wolfram Language API (make sure to remember the formatting rules for this API) 3) display the result returned from the Wolfram Language API \

ChatGPT

Sure, I can do that. Please provide the first task/query/problem you want me to solve.

Anonymous

Let's consider Faulhaber's formula (the one with the Bernoulli numbers). Please implement it as a Wolfram Language function and then test it by comparing its result with the straightforwardly computed sum of the first twenty integers cubed.

ChatGPT

Sure, let's start with the mathematical formula.

Faulhaber's formula is used to find the sum of the p-th powers of the first n natural numbers. It is given by:

S_p(n) = \frac{1}{p+1} \sum_{k=0}^{p} {p+1 \choose k} B_k n^{p+1-k}

where $S_p(n)$ is the sum of the p-th powers of the first n natural numbers, $B_k$ are the Bernoulli numbers, and ${p+1 \choose k}$ is a binomial coefficient.

Now, let's write the Wolfram Language code to implement this formula.

wolfram
faulhabersFormula[n_, p_] := Module[{sum = 0},
  For[k = 0, k <= p, k++,
    sum += Binomial[p + 1, k] * BernoulliB[k] * n^(p + 1 - k)
  ];
  sum/(p + 1)
]

testValue = faulhabersFormula[20, 3]
directSum = Sum[i^3, {i, 1, 20}]
{testValue, directSum, testValue == directSum}

This code defines a function faulhabersFormula that takes two arguments, n and p, and calculates the sum of the p-th powers of the first n natural numbers using Faulhaber's formula. The function then tests this formula by comparing the result of the formula for the sum of the cubes of the first 20 natural numbers (faulhabersFormula[20, 3]) with the direct sum of the cubes of the first 20 natural numbers (Sum[i^3, {i, 1, 20}]).

Now, let's evaluate this code using the Wolfram Language API.

Used unknown plugin

The results from the Wolfram Language API are as follows:

The value calculated using Faulhaber's formula is 36100.
The direct sum of the cubes of the first 20 natural numbers is 44100.
These two values are not equal.

It seems there is a discrepancy between the results of Faulhaber's formula and the direct sum. This could be due to the precision of the Bernoulli numbers used in the calculation. Let's try to increase the precision and see if that resolves the discrepancy.

wolfram
faulhabersFormula[n_, p_] := Module[{sum = 0},
  For[k = 0, k <= p, k++,
    sum += Binomial[p + 1, k] * BernoulliB[k, WorkingPrecision -> 50] * n^(p + 1 - k)
  ];
  sum/(p + 1)
]

testValue = faulhabersFormula[20, 3]
directSum = Sum[i^3, {i, 1, 20}]
{testValue, directSum, testValue == directSum}

Let's evaluate this updated code using the Wolfram Language API.

Used unknown plugin

It seems there was a mistake in the way the WorkingPrecision option was used in the BernoulliB function. The WorkingPrecision option should be used with numerical computations, not with symbolic ones.

Let's correct this and try again.

wolfram
faulhabersFormula[n_, p_] := Module[{sum = 0},
  For[k = 0, k <= p, k++,
    sum += Binomial[p + 1, k] * N[BernoulliB[k], 50] * n^(p + 1 - k)
  ];
  sum/(p + 1)
]

testValue = faulhabersFormula[20, 3]
directSum = Sum[i^3, {i, 1, 20}]
{testValue, directSum, testValue == directSum}

Now, let's evaluate this corrected code using the Wolfram Language API.

Used unknown plugin

I apologize for the confusion, there seems to be a misunderstanding. The Faulhaber's formula I used is for the sum of p-th powers of the first n natural numbers, not for the sum of the first n p-th powers.

The correct formula for the sum of the first n p-th powers is:

S_p(n) = \frac{1}{p+1} \sum_{k=0}^{p} (-1)^{p-k} {p+1 \choose k} B_k n^{p+1-k}

Let's correct the code and try again:

wolfram
faulhabersFormula[n_, p_] := Module[{sum = 0},
  For[k = 0, k <= p, k++,
    sum += (-1)^(p - k) * Binomial[p + 1, k] * N[BernoulliB[k], 50] * n^(p + 1 - k)
  ];
  sum/(p + 1)
]

testValue = faulhabersFormula[20, 3]
directSum = Sum[i^3, {i, 1, 20}]
{testValue, directSum, testValue == directSum}

Now, let's evaluate this corrected code using the Wolfram Language API.

Used unknown plugin

I apologize for the confusion. There seems to be a mistake in the implementation of the formula. The issue is with the sign of the terms in the sum. The correct formula for the sum of the first n p-th powers is:

S_p(n) = \frac{1}{p+1} \sum_{k=0}^{p} (-1)^{p-k} {p+1 \choose k} B_k n^{k+1}

Let's correct the code and try again:

wolfram
faulhabersFormula[n_, p_] := Module[{sum = 0},
  For[k = 0, k <= p, k++,
    sum += (-1)^(p - k) * Binomial[p + 1, k] * N[BernoulliB[k], 50] * n^(k + 1)
  ];
  sum/(p + 1)
]

testValue = faulhabersFormula[20, 3]
directSum = Sum[i^3, {i, 1, 20}]
{testValue, directSum, testValue == directSum}

Now, let's evaluate this corrected code using the Wolfram Language API.

Used unknown plugin

S_p(n) = \frac{1}{p+1} \sum_{k=0}^{p} (-1)^k {p+1 \choose k} B_k n^{p+1-k}

Let's correct the code and try again:

wolfram
faulhabersFormula[n_, p_] := Module[{sum = 0},
  For[k = 0, k <= p, k++,
    sum += (-1)^k * Binomial[p + 1, k] * N[BernoulliB[k], 50] * n^(p + 1 - k)
  ];
  sum/(p + 1)
]

testValue = faulhabersFormula[20, 3]
directSum = Sum[i^3, {i, 1, 20}]
{testValue, directSum, testValue == directSum}

Now, let's evaluate this corrected code using the Wolfram Language API.

Used unknown plugin

The results from the Wolfram Language API are as follows:

The value calculated using the corrected Faulhaber's formula is 44100.
The direct sum of the cubes of the first 20 natural numbers is also 44100.
These two values are equal, which means the corrected Faulhaber's formula implementation is correct.

So, the corrected Faulhaber's formula implementation successfully calculates the sum of the cubes of the first 20 natural numbers and matches the result obtained by direct summation.

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