# Daniel W. Stroock's A Concise Introduction to Analysis PDF

By Daniel W. Stroock

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M=0 Given > 0, choose n ≥ 1 so that |Sn − S| < for n ≥ n , and conclude first that n −1 ∞ am λm − S ≤ (1 − λ) m=0 |Sm − S|λm + if 0 < λ < 1 m=0 and then that ∞ λ 1 ∞ a m λm = lim m=0 am . 1) 38 1 Analysis on the Real Line This observation, which is usually attributed to Abel, requires that one know ahead of time that ∞ m=0 am converges. Indeed, give an example of a sequence for which the limit on the left exists in R but the series on the right diverges. 1) (−1)m can be useful, use it to show that ∞ = log 21 .

Of course z is not a “real” number, since it is an element of R2 , not R, and for that reason it is called a complex number and the set C of all complex numbers with the additive and multiplicative structure that we have been discussing is called the complex plane. For obvious reasons, the x in z = x + i y is called the real part of z, and, because i is not a real number and for a long time people did not know how to rationalize its existence, the y is called the imaginary part of z. , the distance between (x, y) and the origin).

Remember (cf. 7)) that log x = x1 , and therefore 1 that log (1 − x) = − 1−x for |x| < 1. Using our computation when we derived m the geometric formula, we now see that the mth derivative − ddx m log(1 − x) of − log(1 − x) at x = 0 is (m − 1)! when m ≥ 1. Since log 1 = 0, Taylor’s theorem says that, for each n ≥ 0 there exists a |θx | < |x| such that n − log(1 − x) = m=1 θn+1 xm + x . m n+1 Hence, we have that ∞ log(1 − x) = − m=1 xm for x ∈ (−1, 1). 6) It should be pointed out that there are infinitely differentiable functions for which Taylor’s theorem gives essentially no useful information.