Using known formulae to sum more complex series

Using known formulae to sum more complex series

Utilising Known Formulae to Sum Complex Series

Understanding the Concept

  • Series in mathematics refer to summations of sequences, either finite or infinite. Known formulae are used to simplify the process of summing these sequences.
  • Complex series relate to more intricate sequences, often involving a series of terms that follow a specific pattern or rules which require a deeper understanding to successfully sum.

Application of Formula for Sum of Arithmetic Series

  • An arithmetic series is the sum of a sequence where each term after the first is found by adding a constant difference to the previous term.
  • The formula to find the sum S of a finite arithmetic series S = n/2 [2a + (n-1)d], where a is the first term, n is the number of terms and d is the common difference, can be crucial to simplifying complex series.

Sum of Geometric Series

  • A geometric series is a series where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
  • To find the sum of a geometric series, we use the formula S = a(1 - r^n)/(1 - r), where a is the first term, r is the common ratio, and n is the number of terms.

Sum to Infinity for Geometric Series

  • For an infinite geometric series, when the absolute value of r is less than 1 ( r < 1), the series has a finite value.
  • The sum to infinity of such series can be found using the formula S = a / (1 - r).

Manipulating Series for Easier Computation

  • Some complex series may not be immediately recognisable as arithmetic or geometric. In such cases, manipulating the series to a form more appropriate for the known formulae is needed.
  • Techniques such as pairing terms, separating terms or splitting into partial fractions can help transform a tricky series into a well-recognised form.

The Sigma Notation

  • Sigma (Σ) notation is a powerful tool often used to represent a series. Understanding the Sigma notation is essential in handling more complex series.

Real World Applications

  • Complex series have broad use, ranging from calculating compound interest in finance to understanding population growth in biology and employing Fourier Series in physics and engineering.

Reviewing Work

  • Be sure to consistently check your work while computing complex series. Small errors in calculations can lead to large inaccuracies.
  • Practice utilising the known formulae frequently. Mastery of this topic heavily relies on familiarity with each formula and how to apply them.