91Ó°ÊÓ

In each of Exercises 49-54, use Taylor series to calculate the given limit. $$ \lim _{x \rightarrow 0} \frac{\ln (1+x)-x}{x^{2}} $$

Use a Comparison Test to determine whether the given series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{n+2}{2 n^{3 / 2}+3} $$

Comprises only even powers \((x-c)^{2 n} .\) Make the substitution \(t=(x-c)^{2},\) find the interval of convergence of the series in \(t,\) and use it to find the interval of convergence of the original series. $$ \sum_{n=0}^{\infty} 4^{n}(2 x-1)^{2 n} $$

In Each of Exercises \(51-56,\) the partial sum \(S_{N}=\sum_{n=1}^{N} a_{n}\) of an infinite series \(\sum_{n=1}^{\infty} a_{n}\) is given. Determine the value of the infinite series. $$ S_{N}=2-1 / N^{2} $$

An equation is given that expresses the value of an alternating series. For the given \(d\), use the Alternating Series Test to determine a partial sum that is within \(5 \times 10^{-(d+1)}\) of the value of the infinite series. Verify that the asserted accuracy is achieved. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{n}=\ln (2) \quad d=2 $$

Find the sum of the given series in closed form. State the radius of convergence \(R\). \(\sum_{n=1}^{\infty} n x^{2 n+1}\)

In each of Exercises 49-54, use Taylor series to calculate the given limit. $$ \lim _{x \rightarrow 0} \frac{\arctan (x)-x}{x^{3}} $$

An equation is given that expresses the value of an alternating series. For the given \(d\), use the Alternating Series Test to determine a partial sum that is within \(5 \times 10^{-(d+1)}\) of the value of the infinite series. Verify that the asserted accuracy is achieved. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{4}{2 n-1}=\pi \quad d=2 $$

Use known facts about \(p\) -series to determine whether the given series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{n^{3}+1}{n^{4}} $$

Find the sum of the given series in closed form. State the radius of convergence \(R\). \(\sum_{n=0}^{\infty} n(x+2)^{n}\)