Sums of series

• Sums of series is a major mathematical concept that involves the summing of the elements of a sequence.

• Two basic types of series are arithmetic series and geometric series. The sum formula for each differs depending on the type of series.

• Arithmetic series have a constant difference (common difference) between each term. Their sum can be calculated using the formula: S = n/2(a + l), where S is the sum of the series, n is the number of terms, a is the first term, and l is the last term.

• Geometric series have a common ratio between successive terms. The sum of a finite geometric series can be calculated using the formula: S = a(1 - r^n) / (1 - r), where S is the sum of the series, n is the number of terms, a is the first term, and r is the common ratio.

• The concept of convergence is key in series. An infinite series is said to converge if the partial sums of the series approach a certain finite limit as the number of terms increases.

• In contrast, if the series does not approach any finite limit, it is said to diverge. Understanding where series converge and where they diverge is crucial in this topic.

• ‘Sigma notation’, represented by Σ, is commonly used for convenience in writing long series. Knowing how to manipulate and evaluate series in this notation is vital.

• To evaluate the sum of a series presented in sigma notation, identify the general term, the lower limit and the upper limit within the given notation.

• The series sum rules must be mastered: the sum in a series can be split into the sum of several series (the term-by-term addition rule) and a constant multiple inside a series sum can be moved outside (the scalar multiple rule).

• Understanding these basic principles and rules associated with the sums of series forms a foundation for more advanced mathematical concepts related to calculus, number theory and complex numbers.

• Practice and solve problems involving sums of series to reinforce theoretical understanding and improve problem-solving skills.

• Work on a variety of problems, from simple to complex, to gain a comprehensive understanding of the topic and to be able to handle any question on this topic in the test.