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Chapter 9: Problem 97
Using the Ratio Test, it is determined that an alternating series converges. Does the series converge conditionally or absolutely? Explain.
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the terms of a series \(\sum_{n=1}^{\infty} a_{n}\) are defined recursively. Determine the convergence or divergence of the series. Explain your reasoning. \(a_{1}=\frac{1}{2}, a_{n+1}=\frac{4 n-1}{3 n+2} a_{n}\)
Verify that the Ratio Test is inconclusive for the \(p\) -series. $$\sum_{n=1}^{\infty} \frac{1}{n^{4}}$$
Approximating an Integral In Exercises \(63-70\) , use a power series to approximate the value of the integral with an error of less than \(0.0001 .\) (In Exercises 65 and \(67,\) assume that the integrand is defined as 1 when \(x=0 .\) $$ \int_{0}^{1 / 2} \frac{\arctan x}{x} d x $$
Finding a Taylor Series In Exercises \(1-12,\) use the definition of Taylor series to find the Taylor series, centered at \(c,\) for the function. $$ f(x)=\ln \left(x^{2}+1\right), \quad c=0 $$
Find the values of \(x\) for which the series converges. $$\sum_{n=0}^{\infty} \frac{(x+1)^{n}}{n !}$$
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