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Euler treated these two as special cases of the more general sequence 1 − 2 n + 3 n − 4 n + ..., where n = 1 and n = 0 respectively. This line of research extended his work on the Basel problem and leading towards the functional equations of what are now known as the Dirichlet eta function and the Riemann zeta function .
The partial sums of the series 1 + 2 + 3 + 4 + 5 + 6 + ⋯ are 1, 3, 6, 10, 15, etc.The nth partial sum is given by a simple formula: = = (+). This equation was known ...
In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, [1] is an interpolation polynomial for a given set of data points. The Newton polynomial is sometimes called Newton's divided differences interpolation polynomial because the coefficients of the polynomial are calculated using Newton's ...
In mathematics, for a sequence of complex numbers a 1, a 2, a 3, ... the infinite product ∏ n = 1 ∞ a n = a 1 a 2 a 3 ⋯ {\displaystyle \prod _{n=1}^{\infty }a_{n}=a_{1}a_{2}a_{3}\cdots } is defined to be the limit of the partial products a 1 a 2 ... a n as n increases without bound.
The original use of interpolation polynomials was to approximate values of important transcendental functions such as natural logarithm and trigonometric functions.Starting with a few accurately computed data points, the corresponding interpolation polynomial will approximate the function at an arbitrary nearby point.
The n th partial sum S n is the sum of the first n terms of the sequence; that is, S n = a 1 + a 2 + ⋯ + a n = ∑ k = 1 n a k . {\displaystyle S_{n}=a_{1}+a_{2}+\cdots +a_{n}=\sum _{k=1}^{n}a_{k}.}
For instance, in the above examples, the integer 3 can be partitioned into two parts as 2+1 only. Thus, there is only one monomial in B 3,2. However, the integer 6 can be partitioned into two parts as 5+1, 4+2, and 3+3. Thus, there are three monomials in B 6,2. Indeed, the subscripts of the variables in a monomial are the same as those given by ...
Göbel's sequence can be generalized to kth powers by = + + +. The least indices at which the k-Göbel sequences assume a non-integral value are . 43, 89, 97, 214, 19, 239, 37, 79, 83, 239, ...