(x+y)+(x^2+xy+y^2)+(x^3+x^2y+xy^2+y^3) Upto Infinity

Jasonbradley

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Jun 17, 2024

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The Infinite Sum of Polynomial Powers: Exploring (x+y) + (x^2 + xy + y^2) + (x^3 + x^2y + xy^2 + y^3) + ...

This intriguing series presents an interesting challenge in mathematics. It's a combination of geometric and power series elements. Let's break down how to analyze this infinite sum and explore its potential convergence.

Understanding the Pattern

First, notice that each term in the series is a polynomial:

• Term 1: (x + y)
• Term 2: (x^2 + xy + y^2)
• Term 3: (x^3 + x^2y + xy^2 + y^3)

The pattern is clear: each term is the sum of all possible terms with powers of x and y that add up to the term number.

Expressing the Series Compactly

To work with this series more easily, we can express it using summation notation:

∑_(n=1)^∞ (∑_(k=0)^n (x^(n-k) * y^k)) 

• Outer Summation: The outer sum runs from n = 1 to infinity, representing each term in the series.
• Inner Summation: The inner sum runs from k = 0 to n, generating the individual terms within each polynomial.

Exploring Convergence

The convergence of this infinite series depends critically on the values of x and y:

• Case 1: |x| < 1 and |y| < 1 In this case, each term of the inner summation tends to zero as n approaches infinity. This is because both x and y are raised to increasingly high powers, making the terms progressively smaller. Therefore, the entire series converges.

• Case 2: |x| ≥ 1 or |y| ≥ 1 If either x or y is greater than or equal to 1 in absolute value, the terms of the series do not necessarily approach zero as n increases. This means the series diverges (does not converge to a finite value).

Finding the Sum (for Convergent Cases)

When the series converges, we can find its sum by manipulating the series and using geometric series properties:

1. Rearrange the terms: Notice that the series can be rewritten as:

   (x + x^2 + x^3 + ...) + (y + xy + x^2y + ...) + (y^2 + xy^2 + x^2y^2 + ...) + ...
   

2. Geometric Series: Each of the infinite series in parentheses is a geometric series. For example:

   x + x^2 + x^3 + ... = x(1 + x + x^2 + ...) = x/(1-x)
   

3. Apply Geometric Series Formula: Using the formula for the sum of an infinite geometric series (a/(1-r)), we can find the sum of each infinite series.

4. Combine Terms: Finally, sum up the results for all the infinite series to find the total sum of the original series.

Conclusion

The infinite series (x+y) + (x^2 + xy + y^2) + (x^3 + x^2y + xy^2 + y^3) + ... is a fascinating combination of polynomial and geometric series elements. It converges when both x and y are strictly less than 1 in absolute value. By carefully rearranging terms and applying the geometric series formula, we can determine the sum of this series for convergent cases. This exploration highlights the intricate interplay between different mathematical concepts and the importance of understanding convergence when dealing with infinite sums.