![]() It is a convenient and simple form of shorthand used to give a concise expression for a sum of the values of a variable. Summation is the addition of a sequence of numbers. Theorem 7 (Alternating Series test): If, and, the ‘alternating series’ will converge.Theorem 6 (Conditional Convergence test): A series is said to be conditionally convergent if the series diverges but the series converges.Theorem 5 (Absolute Convergence test): A series is said to be absolutely convergent if the series converges.Theorem 4 (Root test): Suppose that the following limit exists.Theorem 3 (Ratio test): Suppose that the following limit exists.Theorem 2 (Limit Comparison test): Let and, and suppose that.(2) The convergence of implies the convergence of (1) The convergence of implies the convergence of ![]() Theorem 1 (Comparison test): Suppose for for some k.If and be convergent series then if for all n N then.if or its limit doesn’t exist or is plus or minus infinity) then the series is also called divergent. Likewise, if the sequence of partial sums is a divergent sequence (i.e. its limit exists and is finite) then the series is also called convergent i.e. If the sequence of partial sums is a convergent sequence (i.e. The series can be finite or infinte.ĥ + 2 + (-1) + (-4) is a finite series obtained by subtracting 3 from the previous number.ġ + 1 + 2 + 3 + 5 is an infinite series called the Fibonacci series obtained from the ![]() A series is represented by ‘S’ or the Greek symbol. If the sequence is the expression is called the series associated with it. If and are convergent sequences, the following properties hold:Ī series is simply the sum of the various terms of a sequence. This theorem is a useful theorem giving the convergence/divergence and value (for when it’s convergent) of a sequence that arises on occasion. Theorem 5: The sequence is convergent if and divergent for Theorem 4: If and the function f is continuous at L, then Note that in order for this theorem to hold the limit MUST be zero and it won’t work for a sequence whose limit is not zero. Theorem 2 (Squeeze Theorem): If for all n > N for some N and then ![]() Theorem 1: Given the sequence if we have a function f(x) such that f(n) = and then This theorem is basically telling us that we take the limits of sequences much like we take the limit of functions. Mathematics | Rings, Integral domains and Fields.Mathematics | Independent Sets, Covering and Matching.Mathematics | Sequence, Series and Summations.Mathematics | Generating Functions – Set 2.Discrete Maths | Generating Functions-Introduction and Prerequisites.Mathematics | Total number of possible functions.Mathematics | Classes (Injective, surjective, Bijective) of Functions.Number of possible Equivalence Relations on a finite set.Mathematics | Closure of Relations and Equivalence Relations.Mathematics | Representations of Matrices and Graphs in Relations.Discrete Mathematics | Representing Relations.Mathematics | Introduction and types of Relations.Mathematics | Partial Orders and Lattices.Mathematics | Power Set and its Properties.Inclusion-Exclusion and its various Applications.Mathematics | Set Operations (Set theory).Mathematics | Introduction of Set theory.ISRO CS Syllabus for Scientist/Engineer Exam.ISRO CS Original Papers and Official Keys.GATE CS Original Papers and Official Keys.
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