Series
Arithmetic Series
- An arithmetic series is a sequence of numbers in which the difference between any two successive numbers is constant. This constant is known as the ‘common difference’.
- The sum of the first ‘n’ terms of an arithmetic series can be calculated using the formula: S = 0.5n[a + l] or S = 0.5n[2a + (n-1)d], where ‘S’ is the sum, ‘n’ is the number of terms, ‘a’ is the first term, ‘l’ is the last term, and ‘d’ is the common difference.
- The ‘nth’ term of an arithmetic series can be found using the formula: a + (n - 1)d.
Geometric Series
- A geometric series is a sequence of numbers in which the ratio of any two successive numbers is constant. This constant is known as the ‘common ratio’.
- The sum of the first ‘n’ terms of a geometric series can be calculated using the formula: S = a [1 - r^n] / 1 - r, for r≠1, where ‘S’ is the sum, ‘n’ is the number of terms, ‘a’ is the first term, and ‘r’ is the common ratio.
- The sum to infinity, ‘S’, of a geometric series can be found using the formula: S = a / (1 - r), and is only valid if abs(r) < 1.
- The ‘nth’ term of a geometric series can be found using the formula: a * r^{n-1}.
Sigma Notation
- Sigma notation is a concise way of denoting the sum of a series, especially when the series is long.
- The Greek uppercase letter Sigma (Σ) is used to represent the sum.
- A series from ‘a’ to ‘b’ of some function ‘f(i)’ of ‘i’ where ‘a’ and ‘b’ are the lower and upper limits of summation and would be notated as Σ from ‘a’ to ‘b’ of ‘f(i)’.
Convergence and Divergence of Series
- A series is said to converge if the sum of the terms tends towards a finite number as the number of terms approaches infinity.
- If the sum of the terms tends towards infinity, or does not approach any value as the number of terms increases, then the series is said to diverge.
- A necessary condition for a series to converge is that the limit as ‘n’ approaches infinity of the ‘nth’ term equals zero. But this is not a sufficient condition. It means that if the limit is not zero, then the series definitely diverges.
- A geometric series converges if abs(r) < 1 and diverges if abs(r) > 1. If r=1, the sum is simply ‘n’ times the first term.
Binomial Series Expansion
- A binomial series involves the expansion of a power of a binomial theorem (a + b)^n.
- The Binomial Theorem provides a formula for expanding powers of binomials: (a + b)^n = Σ [nCk * a^{n-k} * b^k], where the sum is from ‘k=0’ to ‘n’, and ‘nCk’ is a binomial coefficient.
- The binomial coefficient ‘nCk’ is calculated as ‘n! / [k!(n-k)!]’, where ‘!’ denotes factorial.