- #1

- 7

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[tex]\[ \prod_{n=1}^\infty (1 + x^n)\][/tex]

But is there a closed form for it?

- Thread starter photis
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- #1

- 7

- 0

[tex]\[ \prod_{n=1}^\infty (1 + x^n)\][/tex]

But is there a closed form for it?

- #2

- 42

- 2

Write out the first few terms, [tex](1+x)(1+x^{2})(1+x^{3})(1+x^{4})(1+x^{5})(1+x^{6})(1+x^{7})[/tex]

You can see that the coefficent of the constant term in the expansion will be just 1.

Coefficent of the x term will be 1

Coefficent of the x^2 term will be 1

Coefficent of the x^3 term will be

x^4 will have coefficent 2 because x^4 comes from x^4 and x^3 * x (ie 4 = 4+0 or 3+1)

x^5 is 3, because 5 = 5+0 or 5=4+1 or 5=3+2

x^6 is 4 because 6=6+0 or 6=5+1 or 6=4+2 or 6 = 3+2+1)

See how it works? The coefficent of x^n will be the number of ways you can express n as a sum of unique integers. The closed form for this I don't think exists.

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