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formal calculation wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 examples toggle examples subsection 1 1 formal power series 1 2 symbol manipulation 1 2 1 differential equations 1 2 2 cross product 2 see also 3 references toggle the table of contents formal calculation 3 languages বাংলা ido português edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia this article needs more citations please help improve this article by adding citations to reliable sources unsourced material may be challenged and removed find sources formal calculation news newspapers books scholar jstor april 2026 learn how and when to remove this message in mathematical logic a formal calculation or formal operation is a type of mathematical calculation often involving power series that is carried out purely algebraically while disregarding questions of convergence 1 the expressions are manipulated according to algebraic rules without requiring that the underlying series or operations necessarily converge in the analytical sense this approach is useful when the structure of the calculation is more important than its analytical properties examples edit formal calculations can lead to results that are wrong in one context but correct in another context the equation n 0 q n 1 1 q displaystyle sum _ n 0 infty q n frac 1 1 q holds if q has an absolute value less than 1 ignoring this restriction and substituting q 2 to leads to n 0 2 n 1 displaystyle sum _ n 0 infty 2 n 1 substituting q 2 into the proof of the first equation yields a formal calculation that produces the last equation but it is wrong over the real numbers since the series does not converge however in other contexts e g working with 2 adic numbers or with integers modulo a power of 2 the series does converge the formal calculation implies that the last equation must be valid in those contexts another example is obtained by substituting q 1 the resulting series 1 1 1 1 is divergent over the real and the p adic numbers but a value can be assigned to it with an alternative method of summation such as cesàro summation the resulting value 1 2 is the same as that obtained by the formal computation formal power series edit formal power series is a concept that adopts the form of power series from real analysis the word formal indicates that the series need not converge in mathematics and especially in algebra a formal series is an infinite sum that is considered independently from any notion of convergence and can be manipulated with algebraic operations on series addition subtraction multiplication division partial sums etc a formal power series is a special kind of formal series which may be viewed as a generalization of a polynomial where the number of terms is allowed to be infinite with no requirements of convergence thus the series may no longer represent a function of its variable merely a formal sequence of coefficients in contrast to a power series which defines a function by taking numerical values for the variable within a radius of convergence in a formal power series the powers of the variable are used only as position holders for the coefficients so that the coefficient of x 5 displaystyle displaystyle x 5 is the fifth term in the sequence in combinatorics the method of generating functions uses formal power series to represent numerical sequences and multisets for instance allowing concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved more generally formal power series can include series with any finite or countable number of variables and with coefficients in an arbitrary ring rings of formal power series are complete local rings which supports calculus like methods in the purely algebraic framework of algebraic geometry and commutative algebra they are analogous to p adic integers which can be defined as formal series of the powers of p symbol manipulation edit differential equations edit see also leibniz s notation to solve the differential equation d y d x y 2 displaystyle frac dy dx y 2 these symbols can be treated as ordinary algebraic symbols and without giving any justification regarding the validity of this step we take reciprocals of both sides d x d y 1 y 2 displaystyle frac dx dy frac 1 y 2 a simple antiderivative x 1 y c displaystyle x frac 1 y c y 1 c x displaystyle y frac 1 c x because this is a formal calculation it is acceptable to let c displaystyle c infty and obtain another solution y 1 x 1 0 displaystyle y frac 1 infty x frac 1 infty 0 the final solutions can be checked to confirm that they solve the equation cross product edit see also cross product computing the cross product can be expressed as the following determinant a b i j k a 1 a 2 a 3 b 1 b 2 b 3 displaystyle mathbf a times b begin vmatrix mathbf i mathbf j mathbf k a_ 1 a_ 2 a_ 3 b_ 1 b_ 2 b_ 3 end vmatrix where i j k displaystyle mathbf i mathbf j mathbf k is a positively oriented orthonormal basis of a three dimensional oriented euclidean vector space while a 1 a 2 a 3 b 1 b 2 b 3 displaystyle a_ 1 a_ 2 a_ 3 b_ 1 b_ 2 b_ 3 are scalars such that a a 1 i a 2 j a 3 k displaystyle mathbf a a_ 1 mathbf i a_ 2 mathbf j a_ 3 mathbf k and similar for b displaystyle mathbf b see also edit formal power series mathematical logic references edit nelson david 2008 the penguin dictionary of mathematics 4th ed london penguin uk isbn 978 0141920870 retrieved from https en wikipedia org w index php title formal_calculation oldid 1348571546 category mathematical logic hidden categories articles needing additional references from april 2026 all articles needing additional references this page was last edited on 13 april 2026 at 09 03 utc page was rendered with parsoid text is available under the creative commons attribution sharealike 4 0 license additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about wikipedia disclaimers contact wikipedia legal safety contacts code of conduct developers statistics cookie statement mobile view search search toggle the table of contents formal calculation 3 languages add topic