Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/36

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12

DIFFERENTIAL CALCULUS

Writing (B) and (C) in the forms

$\scriptstyle {{\frac {2}{3}}-S_{2n}={\frac {1}{3\cdot 2^{2n-1}}},\quad S_{2n+1}-{\frac {2}{3}}={\frac {1}{3\cdot 2^{2n}}}}$ ,

we have

\scriptstyle {{\underset {n=\infty }{\operatorname {limit} }}\left({\frac {2}{3}}-S_{2n}\right)={\underset {n=\infty }{\operatorname {limit} }}{\frac {1}{3\cdot 2^{2n-1}}}=0}

,

and

\scriptstyle {{\underset {n=\infty }{\operatorname {limit} }}\left(S_{2n+1}-{\frac {2}{3}}\right)={\underset {n=\infty }{\operatorname {limit} }}{\frac {1}{3\cdot 2^{2n}}}=0}

.

Hence, by definition of the limit of a variable, it is seen that both $\scriptstyle {S_{2n}}$ and $\scriptstyle {S_{2n+1}}$ are variables approaching $\scriptstyle {\frac {2}{3}}$ as a limit as the number of terms increases without limit.

Summing up the first two, three, four, etc., terms of (A), the sums are found by (B) and (C) to be alternately less and greater than $\scriptstyle {\frac {2}{3}}$ , illustrating the case when the variable, in this case the sum of the terms of (A), is alternately less and greater than its limit.

In the examples shown the variable never reaches its limit. This is not by any means always the case, for from the definition of the limit of a variable it is clear that the essence of the definition is simply that the numerical value of the difference between the variable and its limit shall ultimately become and remain less than any positive number we may choose, however small.

(4) As an example illustrating the fact that the variable may reach its limit, consider the following. Let a series of regular polygons be inscribed in a circle, the number of sides increasing indefinitely. Choosing any one of these, construct the circumscribed polygon whose sides touch the circle at the vertices of the inscribed polygon. Let $\scriptstyle {p_{n}}$ and $\scriptstyle {P_{n}}$ be the perimeters of the inscribed and circumscribed polygons of $\scriptstyle {n}$ sides, and $\scriptstyle {C}$ the circumference of the circle, and suppose the values of a variable $\scriptstyle {x}$ to be as follows:

$\scriptstyle {P_{n},\quad p_{n+1},\quad C,\quad P_{n+1},\quad p_{n+2},\quad C,\quad P_{n+2},\quad }$ etc.

Then, evidently,

\scriptstyle {{\underset {n=\infty }{\operatorname {limit} }}\;x=C}

,

and the limit is reached by the variable, every third value of the variable being C.

14. Division by zero excluded. $\scriptstyle {\frac {0}{0}}$ is indeterminate. For the quotient of two numbers is that number which multiplied by the divisor will give the dividend. But any number whatever multiplied by zero gives