series

Examples of series in the following topics:

• Summing an Infinite Series

  • A series is the sum of the terms of a sequence.
  • Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely.
  • Infinite sequences and series can either converge or diverge.
  • Working out the properties of the series that converge even if infinitely many terms are non-zero is, therefore, the essence of the study of series.
  • In the following atoms, we will study how to tell whether a series converges or not and how to compute the sum of a series when such a value exists.

• MLA: Series and Lists

• APA: Series and Lists

• Chicago/Turabian: Series and Lists

• MLA: Series and Lists

• Chicago/Turabian: Series and Lists

• Tips for Testing Series

  • When testing the convergence of a series, you should remember that there is no single convergence test which works for all series.
  • It is up to you to guess and pick the right test for a given series.
  • But if the integral diverges, then the series does so as well.
  • Direct comparison test: If the series $\sum_{n=1}^\infty b_n$ is an absolutely convergent series and $\left |a_n \right | \le \left | b_n \right|$ for sufficiently large $n$, then the series $\sum_{n=1}^\infty a_n$ converges absolutely.
  • The integral test applied to the harmonic series.

• Taylor and Maclaurin Series

  • If the Taylor series is centered at zero, then that series is also called a Maclaurin series, named after the Scottish mathematician Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century.
  • A function may not be equal to its Taylor series, even if its Taylor series converges at every point.
  • In the case that $a=0$, the series is also called a Maclaurin series.
  • The Maclaurin series for $(1 − x)^{−1}$ for $\left| x \right| < 1$ is the geometric series: $1+x+x^2+x^3+\cdots\!
  • Identify a Maclaurin series as a special case of a Taylor series

• Resisitors in Series

  • shows resistors in series connected to a voltage source.
  • Using Ohm's Law to Calculate Voltage Changes in Resistors in Series
  • $RN (series) = R_1 + R_2 + R_3 + ... + R_N.$
  • A brief introduction to series circuit and series circuit analysis, including Kirchhoff's Current Law (KCL) and Kirchhoff's Voltage Law (KVL).
  • Three resistors connected in series to a battery (left) and the equivalent single or series resistance (right).

• Power Series

  • A power series (in one variable) is an infinite series of the form $f(x) = \sum_{n=0}^\infty a_n \left( x-c \right)^n$, where $a_n$ is the coefficient of the $n$th term and $x$ varies around $c$.
  • A power series (in one variable) is an infinite series of the form:
  • This series usually arises as the Taylor series of some known function.
  • can be written as a power series around the center $c=1$ as:
  • In such cases, the power series takes the simpler form