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2. Arithmetico-geometric sequence - Wikipedia

   en.wikipedia.org/wiki/Arithmetico-geometric_sequence

   v. t. e. In mathematics, an arithmetico-geometric sequence is the result of term-by-term multiplication of a geometric progression with the corresponding terms of an arithmetic progression. The n th term of an arithmetico-geometric sequence is the product of the n th term of an arithmetic sequence and the n th term of a geometric sequence. [1]

3. Arithmetic progression - Wikipedia

   en.wikipedia.org/wiki/Arithmetic_progression

   For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common difference of 2. If the initial term of an arithmetic progression is a 1 {\displaystyle a_{1}} and the common difference of successive members is d {\displaystyle d} , then the n {\displaystyle n} -th term of the sequence ( a n {\displaystyle a_{n ...

4. Series (mathematics) - Wikipedia

   en.wikipedia.org/wiki/Series_(mathematics)

   t. e. In mathematics, a series is, roughly speaking, an addition of infinitely many quantities, one after the other. [1] The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating ...

5. Convergence tests - Wikipedia

   en.wikipedia.org/wiki/Convergence_tests

   Raabe–Duhamel's test. Let { an } be a sequence of positive numbers. Define. If. exists there are three possibilities: if L > 1 the series converges (this includes the case L = ∞) if L < 1 the series diverges. and if L = 1 the test is inconclusive. An alternative formulation of this test is as follows.

6. nth-term test - Wikipedia

   en.wikipedia.org/wiki/Nth-term_test

   In mathematics, the nth-term test for divergence[1] is a simple test for the divergence of an infinite series: If or if the limit does not exist, then diverges. Many authors do not name this test or give it a shorter name. [2]

7. Recurrence relation - Wikipedia

   en.wikipedia.org/wiki/Recurrence_relation

   In linear recurrences, the n th term is equated to a linear function of the previous terms. A famous example is the recurrence for the Fibonacci numbers , F n = F n − 1 + F n − 2 {\displaystyle F_{n}=F_{n-1}+F_{n-2}} where the order k {\displaystyle k} is two and the linear function merely adds the two previous terms.

8. Harmonic series (mathematics) - Wikipedia

   en.wikipedia.org/wiki/Harmonic_series_(mathematics)

   Calculus. In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: The first terms of the series sum to approximately , where is the natural logarithm and is the Euler–Mascheroni constant. Because the logarithm has arbitrarily large values, the harmonic series does not have a finite limit: it ...

9. Sequence - Wikipedia

   en.wikipedia.org/wiki/Sequence

   The partial sums of a series are the expressions resulting from replacing the infinity symbol with a finite number, i.e. the Nth partial sum of the series = is the number S N = ∑ n = 1 N a n = a 1 + a 2 + ⋯ + a N . {\displaystyle S_{N}=\sum _{n=1}^{N}a_{n}=a_{1}+a_{2}+\cdots +a_{N}.}