Exercise · ZW-PENTAGONAL-NUMBER-THEOREM-009 · ★★★ Advanced
Use the theorem to show that the coefficient of \(x^{22}\) in \(\prod_{n\ge1}(1-x^n)\) is 1, stating the value of \(k\).
Choose one option.
Show answer
- B. 1, with k=4 and sign \((-1)^4\)
Explanation
With \(k=4\) we get \(4(11)/2=22\) and the sign is \((-1)^4=+1\), so the coefficient is 1. Taking the sign as negative or rejecting 22 as non-pentagonal are typical errors.
Review the concept: Pentagonal number theorem
Euler's pentagonal number theorem states that the product \(\prod_{n\ge1}(1-x^n)\) equals an alternating sum of powers of \(x\) indexed by generalized pentagonal numbers.
\[\prod_{n\ge1}(1-x^n)=\sum_{k=-\infty}^{\infty}(-1)^k x^{k(3k-1)/2}\]
More practice problems: pentagonal number theorem
- ZW-PENTAGONAL-NUMBER-THEOREM-001 What sign does the term with exponent k(3k-1)/2 have when k=1 in the theorem's sum?
- ZW-PENTAGONAL-NUMBER-THEOREM-002 What generalized pentagonal exponent is obtained with k=2?
- ZW-PENTAGONAL-NUMBER-THEOREM-003 What exponent does k=-1 give in k(3k-1)/2?
- ZW-PENTAGONAL-NUMBER-THEOREM-004 What is the coefficient of x^0 in prod_n≥1(1-x^n)?
- ZW-PENTAGONAL-NUMBER-THEOREM-005 What is the sign of the x^1 term in the expansion of the product?
- ZW-PENTAGONAL-NUMBER-THEOREM-006 Compute the coefficient of x^7 in prod_n≥1(1-x^n) using the pentagonal number theorem.
- ZW-PENTAGONAL-NUMBER-THEOREM-007 How many exponents below 10 have a nonzero coefficient in prod_n≥1(1-x^n)?
- ZW-PENTAGONAL-NUMBER-THEOREM-008 Is it true that every exponent m with nonzero coefficient in prod_n≥1(1-x^n) is even?
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