216 Answers to Queries. 
395.—Infinite and other Series.—The phrase, “sum of an 
infinite series,” means that limit (if any) to which the sum of a 
finite number of terms approximates constantly and with indefi- 
nite closeness as the number of terms is indefinitely increased. 
An infinite series has no finite sum if no such limit exists. An 
infinite divergent series may exactly resemble a binomial expan- 
sion. But the binomial theorem is only true when the series is 
convergent (see Todhunter’s Algebra, § 519, and the answer by 
G. Ek. B:5 p. 175): 
Hence the series, r—5 + 15—35 + 70-—126 + ..., is not 
=(1+1)?°= a Its sum to z terms can be found, in virtue of 
the fact that the series 1, 5, 15, 35, 70, 126 ... has its third differ- 
ences in arithmetical progression. ‘The sum is + (n + 1) (n+ 3) 
(n2 + 4n + 1) + 48 when z is odd, and —n (n + 2) (n + 4) 
— 48 when zis even. Putting infinite, the sum is + a 
Series whose differences (of any order) have received, I think 
no general name, their general term can always be expressed as a 
sum of multiples of figurate numbers. Thus, the general term of 
the series 75 8, 8, 8, 9; 12, 18, is —— ays | ——s oF 
6n + 4. The treatment of all such series reduces, therefore, to 
that of figurate ones. C. W. C. BaRLow. 
407.—Rule Wanted.—The only rule for solving such a 
problem needs algebraical symbols, and even that is highly 
laborious except in simple cases. By making the arbitrary 
assumption that there was one shilling and one of every coin of 
higher value, but thirty 45 pieces, I found eighty solutions. The 
actual number is reckoned by billions. C.-W. C. Bartow, 
409.—Another Rule Wanted.—A long rod is divided into two 
parts, of which the shorter is 28 ft. Add = of longer piece to 
shorter, and then add = of remainder. The result will be 4 of 
the whole. 
+, of remainder is not fof lonfen piece, — —- 
+ , together with a is = 
Thus, if 7 of longer piece is added to shorter, the result is 
= of the whole rod, — of longer, and = of shorter. 
It follows that * of longer = 2 of longer and = of shorter. 
33 25 
” 7 ” Ss » . and r6ieet 
8 
a7 A = 16 feet. 
And the longer part is a of 16 feet, = 154 feet. 
The entire rod was, therefore, 182 feet long. The work would 
be much shortened by using algebraical notation. C. W. ANB: 
