Harriot, Thomas - Mss. 6785

356178v

[Commentary:

Sums of some infinite progressions.
The first example

\frac{1}{1} + \frac{2}{2} + \frac{3}{4} + \frac{4}{8} + \frac{5}{16} + . . . = 4

. From the similar examples shown on Add MS 6789 f. , we may assume that Harriot summed this as follows:

\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + . . . = 2

\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + . . . = 1

\frac{1}{4} + \frac{1}{8} + . . . = \frac{1}{2}

. . .

. The sums of these series form a geometric progression

2 + 1 + \frac{1}{2} + \frac{1}{4} + . . . = 4

.
The second example is similar to the first.
The third example

\frac{1}{1} + \frac{3}{3} + \frac{6}{9} + \frac{11}{27} + \frac{20}{81} + . . . = 3 \frac{3}{4}

. This can be rewritten as the sum of two separate series:

\frac{1}{1} + \frac{2}{3} + \frac{4}{9} + \frac{8}{27} + \frac{16}{81} + . . .

. and

\frac{1}{3} + \frac{2}{9} + \frac{3}{27} + \frac{4}{81} + . . .

. The first is a geometric progression whose sum is 3.
The second can be summed as in the first example, to give

\frac{3}{4}

.
Thus the total sum is

3 \frac{3}{4}

.
The fourth example

\frac{2}{1} + \frac{5}{6} + \frac{13}{36} + \frac{35}{216} + . . .

. This can be rewritten as the sum of two separate series:

\frac{1}{1} + \frac{2}{6} + \frac{4}{36} + \frac{9}{216} + . . .

. and

\frac{1}{1} + \frac{3}{6} + \frac{9}{36} + \frac{27}{216} + . . .

. The first is a geometric progression whose sum is 3.
The second is a geomteric progression whose sum is 2.
Thus the total sum is 5.
]