Infinite Series

Excerpts

• Infinite Series

  If in $\phi x$, any function of $x$, the value of $x$ be increased by $h$, or $x + h$ be substituted instead of $x$, the result is denoted by $\phi(x + h)$.

• Infinite Series

  It will happen, however, in many functions, that one or more values can be given to $x$ for which it is impossible to expand $f(x + h)$ without introducing negative or fractional powers.

• Infinite Series

  As the notion of a series which has no end of its terms, may be new to the student, we will now proceed to show that there may be series so constructed, that the addition of any number of their terms, however great, will always give a result less than some determinate quantity.

• Infinite Series

  The first two terms of this series may be obtained by dividing $1 - x^{2}$ by $1 - x$; the first three by dividing $1 - x^{3}$ by $1 - x$; and the first $n$ terms by dividing $1 - x^{n}$ by $1 - x$.

• Infinite Series

  Hence by taking $n$ sufficiently great, $\dfrac{1 - x^{n}}{1 - x}$ or $\dfrac{1}{1 - x} - \dfrac{x^{n}}{1 - x}$ may be brought as near to $\dfrac{1}{1 - x}$ as we please, than which, however, it must always be less, since $\dfrac{x^{n}}{1 - x}$ can never entirely vanish, whatever value $n$ may have, and therefore there is always something subtracted from $\dfrac{1}{1 - x}$.

• Infinite Series

  Thus it generally happens that $x^{2} - 10x + 40$ is greater than $15$, with the exception only of the case where $x = 5$. It is generally true that a line which meets a circle in a given point meets it again, with the exception only of the tangent.

Equations

• Infinite Series

  \phi x + ph + qh^{2} + rh^{3} + \etc.,\quad \textit{ad infinitum}

  A function of x + h can generally be expanded as an infinite series in whole, positive powers of h, with the constant term phi x.

• Infinite Series

  \phi(x + h) - \phi x

  The increment of phi x is the difference between phi(x + h) and phi x, negative when phi(x + h) is less than phi x.

• Infinite Series

  \phi(x + h) < \phi x

  The increment of phi x is negative exactly when phi(x + h) is less than phi x.

• Infinite Series

  \frac{1}{1 - x} = 1 + x + x^{2} + x^{3} + \etc.,\quad\textit{ad infinitum}

  For x less than unity, the infinite geometric series 1 + x + x^2 + ... can be brought as near as we please to 1/(1 - x), so the series is said to equal it.

Problems

No exercises in this chapter.