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Finding Intervals of Convergence In Exercises 49–52, find the intervals of convergence of (a)
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Calculus (MindTap Course List)
- Absolute convergence and the ratio and root Testsarrow_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_forwardSummation of n = 0 to infinity of (x3n) / (n!) Find radius and interval of convergence.arrow_forward
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- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage