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The document discusses mathematical formulas and proofs. It contains: 1) Formulas for polynomials and series expansions using binomial coefficients. 2) A claim and proof about the series expansion of (1-x)-3 using binomial theorem. 3) Notations showing equivalence and equality of expressions.

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Mathtype f ( x) = an x n + an ?1 x n ?1 + L , an ¡Ù 0 an ?1 n ?1 = (x + n x + L), exist an = ( x m + bm ?1 x m ?1 + L) ³Ìʽ¾ŽÝ‹Æ÷ claim: ¡Æ hx n 1 3 n¡Ý0 n = (1 ? x ) 13 + 23 +L + n3 =? A=B A= A1 = A2 = L =B B= B1 = B2 = L =A (1 + x ) 2 =1+2x+ x 2 1 3 1 1 1 2 2 2 Pf: (1 ? x ) = (1 ? x ) (1 ? x ) (1 ? x ) = (1+x+x +L ) (1+x+x +L ) (1+x+x +L ) = ¡Æxei ¡Ý 0 e1 x e2 x e3 = ¡Æx ei ¡Ý 0 e1 + e2 + e3 = ¡Æ (# of sols of e1 + e2 + e3 =n) x n n¡Ý 0 On the other hand,we have by newton¡¯s binomial theorem ? ?3 ? ? n + 2? n ? 2? n ) = ( x ? 1) ?3 = ¡Æ ? ? (? x ) n = ¡Æ ? ? x ? ? = ¡Æ n ¡­.(1) 1 3 h xn ( 1? x n¡Ý 0 ?n? n¡Ý 0 ? 2 ? ? 1 ? n¡Ý0

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  • 1. Mathtype f ( x) = an x n + an ?1 x n ?1 + L , an ¡Ù 0 an ?1 n ?1 = (x + n x + L), exist an = ( x m + bm ?1 x m ?1 + L) ³Ìʽ¾ŽÝ‹Æ÷ claim: ¡Æ hx n 1 3 n¡Ý0 n = (1 ? x ) 13 + 23 +L + n3 =? A=B A= A1 = A2 = L =B B= B1 = B2 = L =A (1 + x ) 2 =1+2x+ x 2 1 3 1 1 1 2 2 2 Pf: (1 ? x ) = (1 ? x ) (1 ? x ) (1 ? x ) = (1+x+x +L ) (1+x+x +L ) (1+x+x +L ) = ¡Æxei ¡Ý 0 e1 x e2 x e3 = ¡Æx ei ¡Ý 0 e1 + e2 + e3 = ¡Æ (# of sols of e1 + e2 + e3 =n) x n n¡Ý 0 On the other hand,we have by newton¡¯s binomial theorem ? ?3 ? ? n + 2? n ? 2? n ) = ( x ? 1) ?3 = ¡Æ ? ? (? x ) n = ¡Æ ? ? x ? ? = ¡Æ n ¡­.(1) 1 3 h xn ( 1? x n¡Ý 0 ?n? n¡Ý 0 ? 2 ? ? 1 ? n¡Ý0