91Ó°ÊÓ

In Exercises \(47-50\), determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If \(\sum_{n=1}^{\infty} a_{n}\) and \(\sum_{n=1}^{\infty} b_{n}\) converge absolutely, then \(\sum_{n=1}^{\infty}\left(a_{n}+b_{n}\right)\) converges absolutely.

Use a power series to obtain an approximation of the definite integral to four decimal places of accuracy. \(\int_{0}^{0.5} x \cos x^{3} d x\)

In Exercises \(57-62\), find the sum of the given series. (Hint: Each series is the Maclaurin series of a function evaluated at an appropriate point.) \(\sum_{n=1}^{\infty}(-1)^{n-1} \frac{1}{n}\)

An infinite sequence of nested squares is constructed as follows: Starting with a square with a side of length 2 , each square in the sequence is constructed from the preceding square by drawing line segments connecting the midpoints of the sides of the square. Find the sum of the areas of all the squares in the sequence.

An infinite sequence of nested equilateral triangles and circles is constructed as follows: Beginning with an equilateral triangle with a side of length 1 , inscribe a circle followed by a triangle, followed by a circle, and so on, ad infinitum. Find the total area of the shaded regions.