This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1896 Excerpt: ... series on the right, we obtain the form in which the theorem is usually given, namely (a + b)'-a" + wa-1 b +-(" a""3 6 +..... + T-4---a-rbr+... + b r n--r The series on the right is called the expansion of (a + b) 254. Proof by Induction. The Binomial Theorem may also be proved by induction, as follows. 255. General ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1896 Excerpt: ... series on the right, we obtain the form in which the theorem is usually given, namely (a + b)'-a" + wa-1 b +-(" a""3 6 +..... + T-4---a-rbr+... + b r n--r The series on the right is called the expansion of (a + b) 254. Proof by Induction. The Binomial Theorem may also be proved by induction, as follows. 255. General term. By the preceding articles we see that any term of the expansion of (a + b)" by the Binomial Theorem will be found by giving a suitable value to r in In T-h--nr. r vn--r On this account the above is called the general term of the series. It should be noticed that the term is the (r + l)th term from the beginning. See Note Art. 244. 256. Coefficients of terms equidistant respectively from the beginning and the end are equal. In the expansion of (a + b)n by the Binomial Theorem, the (r + l)th term from the beginning and the (r + l)th term from the end are respectively J0r. a'-'V and (? _, . orb"-. But nGr = nG _T. Art. 245. Hence, in the expansion of (a + b)n, the coefficients of any two terms equidistant respectively from the beginning and the end are equal. This result follows, however, at once from the fact that (a + b)n, and therefore also its expansion, would be unaltered by an interchange of the letters a and b; and hence the coefficient of ab must be equal to the coefficient of b' a." 257. If, in the formula of Art. 253, we put a = 1 and b = x, we have (1 + x)' = l + nx + n(-1K!l+... + T-af+...+x v' 1.2 W n--r This is the most simple form of the Binomial Theorem, and the one which is generally employed. The above form includes all possible cases: if, for example, we want to find (a + b)" by means of it, we have (a+6)-=a(l4)n=a-(l +. L, b, n(n-l)/b J.. m 258. Greatest term ...
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