Question

What does it mean for a sequence to converge? What does it mean for a series to converge? What does it mean for a series to converge absolutely? If a 2 Hint: There is no antiderivative of the standard normal curve, even of its easier-looking avatar e , that is expressible using elementary functions. But how then do we know there is any antiderivative of this function to speak of at all?! 3 To quote Lord Kelvin, as does Michael Spivak on page 74 of Calculus on Manifolds, "A mathematician is one to whom that is as obvious as that twice two makes four is to you."

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Answer 1

Convergence of a sequence A sequence converges when it keeps getting closere and closer to a certain valud. example: t The terms of it are t, and so on. and that sequence converges to , because the terms get closer and closer to o. Let us considere a sequence . Now let ut converge to a limit to L. I definition A sequence ay is said to be convergent to a limit L of given any real positive numbere & theree exists an integer In such that for all min, 1 XM-Lker I convergence of a series A series na the sum of the terms of an infinite sequence of numberes. Given an infinite sequence (aigas, ...) the nth partial sum so is the bum of firest in terms of the sequence & A That is, Sn = With

A series is convergent if the sequence of its partial sump (s. 6 ...) tends to a limit that means the partial sume become closere and closere to a given numbere when the number of thanks of theire terems increases. Definition - go A series converges if that there exists a numbere e such that for every arbitrary small positive number &, there I in a (sufficiently large ) intergere N sich that fore and all n SN, Iso-else. If the series is convergent, the e (necessarily unique is called of the series number the sum

absolute convergence & for any sequence şaq, a, az, ...} ans lang fore all on. Therefore, 3 ans Slail. This means it Elan I convergence, then I am also convergence (but not vice ont versea ns hagues Definition - If the series then the series Slan 7 San convergeg is absolutele convergent. eg:- The power series of the exponential function is absolutely convergent. everywhere