Problem 6
Find the first five terms of the sequence of partial sums. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n !} $$
Problem 6
In Exercises \(3-6,\) find the radius of convergence of the power series. $$ \sum_{n=0}^{\infty} \frac{(2 n) ! x^{2 n}}{n !} $$
Problem 6
Use the Integral Test to determine the convergence or divergence of the series. $$ \frac{\ln 2}{2}+\frac{\ln 3}{3}+\frac{\ln 4}{4}+\frac{\ln 5}{5}+\frac{\ln 6}{6}+\cdots $$
Problem 6
\mathrm{Conjecture } Consider the function \(f(x)=x^{2} e^{x}\) (a) Find the Maclaurin polynomials \(P_{2}, P_{3},\) and \(P_{4}\) for \(f\). (b) Use a graphing utility to graph \(f, P_{2}, P_{3},\) and \(P_{4}\). (c) Evaluate and compare the values of \(f^{(n)}(0)\) and \(P_{n}^{(n)}(0)\) for \(n=2,3,\) and 4 (d) Use the results in part (c) to make a conjecture about \(f^{(n)}(0)\) and \(P_{n}^{(n)}(0)\)
Problem 6
Find a power series for the function, centered at \(c,\) and determine the interval of convergence. $$ f(x)=\frac{3}{2 x-1}, \quad c=2 $$
Problem 6
Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1} n}{n^{2}+1} $$
Problem 6
Use the definition to find the Taylor series (centered at \(c\) ) for the function. $$ f(x)=e^{x}, \quad c=1 $$
Problem 6
Write the first five terms of the sequence. \(a_{n}=(-1)^{n+1}\left(\frac{2}{n}\right)\)
Problem 7
Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}(n+1)}{\ln (n+1)} $$
Problem 7
In Exercises \(7-14,\) verify that the infinite series diverges. $$ \sum_{n=0}^{\infty} 1000(1.055)^{n} $$
