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Magazine Problem 58
Use series to approximate the definite integral to within the indicated accuracy. $$ \int_{0}^{1} \sin \left(x^{4}\right) d x \quad \text { (four decimal places) } $$
Problem 58
Find the values of \(x\) for which the series converges. Find the sum of the series for those values of \(x .\) $$ \sum_{n=1}^{\infty}(x+2)^{n} $$
Problem 59
Use series to approximate the definite integral to within the indicated accuracy. $$ \int_{0}^{0.4} \sqrt{1+x^{4}} d x \quad\left(| \text { error } |<5 \times 10^{-6}\right) $$
Problem 59
Find the values of \(x\) for which the series converges. Find the sum of the series for those values of \(x .\) $$ \sum_{n=0}^{\infty} \frac{(x-2)^{n}}{3^{n}} $$
Problem 60
Use series to approximate the definite integral to within the indicated accuracy. $$ \int_{0}^{0.5} x^{2} e^{-x^{2}} d x \quad(| \text { error } |<0.001) $$
Problem 60
Find the values of \(x\) for which the series converges. Find the sum of the series for those values of \(x .\) $$ \sum_{n=0}^{\infty}(-4)^{n}(x-5)^{n} $$
Problem 61
Find the values of \(x\) for which the series converges. Find the sum of the series for those values of \(x .\) $$ \sum_{n=0}^{\infty} \frac{2^{n}}{x^{n}} $$
Problem 61
Use series to evaluate the limit. $$ \lim _{x \rightarrow 0} \frac{x-\ln (1+x)}{x^{2}} $$
Problem 62
Find the values of \(x\) for which the series converges. Find the sum of the series for those values of \(x .\) $$ \sum_{n=0}^{\infty} \frac{\sin ^{n} x}{3^{n}} $$
Problem 62
Use series to evaluate the limit. $$ \lim _{x \rightarrow 0} \frac{1-\cos x}{1+x-e^{x}} $$
