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  2. Euler summation - Wikipedia

    en.wikipedia.org/wiki/Euler_summation

    In the mathematics of convergent and divergent series, Euler summation is a summation method. That is, it is a method for assigning a value to a series, different from the conventional method of taking limits of partial sums. Given a series Σa n, if its Euler transform converges to a sum, then that sum is called the Euler sum of the original ...

  3. Random walk model of consumption - Wikipedia

    en.wikipedia.org/wiki/Random_walk_model_of...

    Robert Hall was the first to derive the effects of rational expectations for consumption. His theory states that if Milton Friedman’s permanent income hypothesis is correct, which in short says current income should be viewed as the sum of permanent income and transitory income and that consumption depends primarily on permanent income, and if consumers have rational expectations, then any ...

  4. Divergent series - Wikipedia

    en.wikipedia.org/wiki/Divergent_series

    Euler summation is essentially an explicit form of analytic continuation. If a power series converges for small complex z and can be analytically continued to the open disk with diameter from ⁠ −1 / q + 1 ⁠ to 1 and is continuous at 1, then its value at q is called the Euler or (E,q) sum of the series Σa n. Euler used it before analytic ...

  5. 1 + 2 + 3 + 4 - 3 + 4 + ... - Wikipedia

    en.wikipedia.org/wiki/Sum_of_natural_numbers

    A summation method that is linear and stable cannot sum the series 1 + 2 + 3 + ⋯ to any finite value. (Stable means that adding a term at the beginning of the series increases the sum by the value of the added term.) This can be seen as follows. If + + + =, then adding 0 to both sides gives

  6. Riemann zeta function - Wikipedia

    en.wikipedia.org/wiki/Riemann_zeta_function

    The values of the Riemann zeta function at even positive integers were computed by Euler. The first of them, ζ(2), provides a solution to the Basel problem. In 1979 Roger Apéry proved the irrationality of ζ. The values at negative integer points, also found by Euler, are rational numbers and play an important role in the theory of modular forms.

  7. Summation - Wikipedia

    en.wikipedia.org/wiki/Summation

    In mathematics, summation is the addition of a sequence of numbers, called addends or summands; the result is their sum or total.Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined.

  8. Euler's formula - Wikipedia

    en.wikipedia.org/wiki/Euler's_formula

    Euler's formula is ubiquitous in mathematics, physics, chemistry, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics". [2] When x = π, Euler's formula may be rewritten as e iπ + 1 = 0 or e iπ = −1, which is known as Euler's identity.

  9. Basel problem - Wikipedia

    en.wikipedia.org/wiki/Basel_problem

    The sum of the series is approximately equal to 1.644934. [3] The Basel problem asks for the exact sum of this series (in closed form), as well as a proof that this sum is correct. Euler found the exact sum to be / and announced this discovery in 1735. His arguments were based on manipulations that were not justified at the time, although he ...