Proof Let f be a positive, continuous, and decreasing function for
converges to S, then the remainder
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Chapter 9 Solutions
Calculus: Early Transcendental Functions
- Real Analysis Show that if {xn} from n=1 to infinity is a convergent sequence in Rd, then there exists N in the positive integers such that XN = XN+1=XN+2 . . . (That is, a sequence in Rd is convergent if and only if all the terms of the sequence are the same from some point on.) Would you please be complete in your explanations of each portion of the answer? I have never seen metric spaces before and am struggling to understand the concept. Thank you.arrow_forwardSummation of n = 0 to infinity of (x3n) / (n!) Find radius and interval of convergence.arrow_forward(Term-by-term Differentiability Theorem). Let fn be differentiable functions defined on an interval A, and assume ∞ n=1 fn(x) converges uniformly to a limit g(x) on A. If there exists a point x0 ∈ [a, b] where ∞ n=1 fn(x0) converges, then the series ∞ n=1 fn(x) converges uniformly to a differentiable function f(x) satisfying f(x) = g(x) on A. In other words, Proof. Apply the stronger form of the Differentiable Limit Theorem (Theorem6.3.3) to the partial sums sk = f1 + f2 + · · · + fk. Observe that Theorem 5.2.4 implies that sk = f1 + f2 + · · · + fk . In the vocabulary of infinite series, the Cauchy Criterion takes the followingform.arrow_forward
- Real Analysis I need a rigorous proof for the following: If 0<=an<=1 (n>0) and if 0<=x<1 then prove that the Series anxn from n=0 to infinity converges and that its sum is not greater than 1/(1-x) Thank you.arrow_forwardTrue or False? Prove your answer! a) Suppose the sequence (xn) does not converge to 0. Then, for every E > 0, infinitely many terms of (xn) lie outside of the interval (−E, E).The claim is:Proof of answer:arrow_forwardinfinity sigma k=0 (-1)^k(k+1)x^k. Use that result to create a power series representation of f(x) = x/(1+x^4)^2arrow_forward
- diverges determine whether the given serics converges absolutely, conditionally, or n+ l 2D the given series is conditionally convergent -@ the series is absolutely convergent b diverges the given seriesarrow_forwardPower series can be used to evaluate certain infinite series. An example is ∞Σ n=1 n/5^(n-1) a) Find a power series representation of 1/(1-x)^2 by differntiating the power series representation of 1/(1-x). What is the interval of convergence for the power series of 1/(1-x)^2? b) Use the your awnser in part (a) to find the sum of ∞Σ n=1 n/5^(n-1). Also, to clear up any confusion, infinity is supposed to be on top and n=1 is on the bottom.arrow_forward
- College AlgebraAlgebraISBN:9781305115545Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage