Of Progressions having an infinite number of terms.

171. Let there be the decreasing progression

:

a b c d e f ...

containing an indefinite number of terms. In the formula

substitute for l its value aq^-1 (Art. 167), and we have

which represents the sum of n terms of the progression. This may be put under the form

Now, since the progression is decreasing, q is a proper fraction; and q" is also a fraction, which diminishes as n increases. Therefore, the greater the number of terms we take, the more will Xq diminish, and consequent

a

1 9

ly the more will the partial sum of these terms approximate to an equality with the first part of S, that is, to

a

1-q

Finally, when n is taken greater than any given number, or n=infinity, then

α

1-q

Xq" will be less than any given

number, or will become equal to 0; and the expression

α

-9

will represent the true value of the sum of all the terms of the series. Whence we may conclude, that the expression

for the sum of the terms of a decreasing progression, in which the number of terms is infinite, is

That is, equal to the first term divided by 1 minus the ratio. This is, properly speaking, the limit to which the partial sums approach, by taking a greater number of terms in the progression. The difference between these sums and

a can become as small as we please, and will only

1-q

become nothing when the number of terms taken is infinite.

We have for the expression of the sum of the terms

The error committed by taking this expression for the value of the sum of the n first terms, is expressed by

QUEST.-165. When the progression is decreasing and the number of terms infinite, what is the value of the sum of the series?

taken for the sum of a certain number of terms, is less in proportion as this number is greater.

2. Again take the progression

172. In the several questions of geometrical progres

sion there are five numbers to be considered:

QUEST-166. How many numbers are considered in geometrical pro

gression? What are they?

173. We shall terminate this subject by the question, To find a mean proportional between any two numbers, as m and n.

Denote the required mean by x. We shall then have (Art. 156),

That is, Multiply the two numbers together, and extract the square root of the product.

1. What is the geometrical mean between the numbers 2 and 8?

Mean√8x2= √16=4 Ans.

2. What is the mean between 4 and 16? 3. What is the mean between 3 and 27 ?

4. What is the mean between 2 and 72 ? 5. What is the mean between 4 and 64?

Ans. 8.

Ans. .9.

Ans. 12.

Ans. 16.

QUEST.-167. How do you find a mean proportional between two

numbers?

THE END.