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The infinite sequence of additions expressed by a series cannot be explicitly performed in sequence in a finite amount of time. However, if the terms and their finite sums belong to a set that has limits, it may be possible to assign a value to a series, called the sum of the series.
For an extreme example, appending a single zero to the front of the series can lead to a different result. [1] One way to remedy this situation, and to constrain the places where zeroes may be inserted, is to keep track of each term in the series by attaching a dependence on some function. [11]
Consider these situations as an example, the two-player zero-sum game pictured at right or above. The order of play proceeds as follows: The first player (red) chooses in secret one of the two actions 1 or 2; the second player (blue), unaware of the first player's choice, chooses in secret one of the three actions A, B or C.
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
A Laurent series is a generalization of the Taylor series, allowing terms with negative exponents; it takes the form = and converges in an annulus. [6] In particular, a Laurent series can be used to examine the behavior of a complex function near a singularity by considering the series expansion on an annulus centered at the singularity.
Zero-sum bias is a cognitive bias towards zero-sum thinking; it is people's tendency to intuitively judge that a situation is zero-sum, even when this is not the case. [4] This bias promotes zero-sum fallacies, false beliefs that situations are zero-sum. Such fallacies can cause other false judgements and poor decisions.
A lump sum could be $10,000, $50,000, $200,000 or any amount that is large given your situation. You might find yourself with a lump sum for any number of reasons. Perhaps you received an inheritance.
Paul Samuelson resolves the paradox [33] by arguing that, even if an entity had infinite resources, the game would never be offered. If the lottery represents an infinite expected gain to the player, then it also represents an infinite expected loss to the host. No one could be observed paying to play the game because it would never be offered.