Summation of series

Summation of series

Basic Concepts

• Series refers to the sum of terms in a sequence. An arithmetic series has a constant difference between terms, whereas a geometric series has a constant ratio.

Arithmetic Series

• The sum of the first n terms of an arithmetic series can be found using the formula:
  • Sₙ = n/2 (2a + (n - 1)d) where a is the first term, and d the common difference.
• If the last term is known, another form is
  • Sₙ = n/2 (a + l) where l represents the last term.

Geometric Series

• The sum of the first n terms of a finite geometric series is given by
  • Sₙ = a (1 - rⁿ) / (1 - r), when the common ratio r is not 1.

• For infinite geometric series, when

  r

  < 1, the sum of the series is

  • S = a / (1 - r)

Sigma Notation

• Sigma (Σ) notation is a shorthand way of expressing a series. If Σaᵢ from i=m to n, then we add up the terms aᵢ for i=m to i=n.

Convergence and Divergence

• A series converges if the sum of its terms approaches a finite value. It diverges if the sum of its terms approaches infinity, or does not approach any value.

• A necessary condition for a geometric series to converge is

  r

  < 1.

Series Tests

• Two prominent series tests are the Comparison Test and the Ratio Test. The comparison test looks at how terms of the series compare to those of a known series, while the ratio test calculates the limit of the ratio of consecutive terms.

Maclaurin and Taylor Series

• A Maclaurin series is a Taylor series expansion of a function about 0. The general formula of a Maclaurin series is
  • f(x) = f(0) + f’(0)x / 1! + f’‘(0)x² / 2! + …
• A Taylor series can provide an approximations for some functions. The general formula is
  • f(x) = f(a) + f’(a)(x - a) / 1! + f’‘(a)(x - a)² / 2! + …