Finding the Radius of Convergence In Exercises 49–52, find the radius of convergence of (a) f(x), (b)
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Calculus: An Applied Approach (MindTap Course List)
- Proof the uniform convergence and differentiationarrow_forwardSeries(a) Find the exact value that X∞n=1√n −√n + 2 converges to, or else concludethat the series diverges.(b) Determine whetherX∞n=1(−1)n· n2 + 2(n − 1)! converges absolutely, converges conditionally, or diverges.arrow_forwardFind the radius of convergence, R, and the interval of convergence, I. infinity signEn=1 xn/n2arrow_forward
- Find the value V of the Riemann sum.arrow_forwardBox answer if it is convergent.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
- Evaluate the Riemann sum for f(x) = x3 − 6x + 2, taking the sample points to be midpoint and a = 0, b = 4, and n = 8b) Find the exact value of ∫03 f(x) dx using Riemann sum c) Find the exact value of ∫03 f(x) dx using the Fundamental Theorem of Calculusarrow_forwardusing the definition of convergence prove that lim n -> infinity. (√(n+1) - √n)=0arrow_forward(Continuous Limit Theorem). Let (fn) be a sequence of functions defined on A ⊆ R that converges uniformly on A to a function f. If each fn is continuous at c ∈ A, then f is continuous at c.arrow_forward
- True or False? Prove your answer! Suppose (xn) does not converge to 0. Then there exists E > 0 such that all except for finitely many terms of (xn) lie outside of the interval (−E, E).The claim is:Proof of answer:arrow_forwardSummation of n = 0 to infinity of (x3n) / (n!) Find radius and interval of convergence.arrow_forwardTrue or false? Justify your answer (a) Every function differentiable infinitely many times at x = 0 is equal to the sum of its Taylor series near x = 0. (c) Suppose that the Taylor series for a function f has an infinite radius of convergence. Then the function is equal to the sum of its Taylor series for every x ∈ R.arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage