91Ó°ÊÓ

Find a recursive definition for the sequence. $$2,4,6,8,10, \dots$$

(a) Show \(r^{\ln n}=n^{\ln r}\) for positive numbers \(n\) and \(r.\) (b) For what values \(r>0\) does \(\sum_{n=1}^{\infty} r^{\ln n}\) converge?

Find a recursive definition for the sequence. $$3,5,9,17,33, \dots$$

Use the alternating series test to decide whether the series converges. $$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{\sqrt{n}}$$

Find the sum of the series. For what values of the variable does the series converge to this sum? $$y-y^{2}+y^{3}-y^{4}+\cdots$$

Use the formula for the sum of a geometric series to find a power series centered at the origin that converges to the expression. For what values does the series converge? $$\frac{3}{1-z / 2}$$

Consider the series \(\sum_{k=1}^{\infty} \frac{1}{k(k+1)}=\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\cdots\) (a) Show that \(\frac{1}{k}-\frac{1}{k+1}=\frac{1}{k(k+1)}\) (b) Use part (a) to find the partial sums \(S_{3}, S_{10},\) and \(S_{n}\) (c) Use part (b) to show that the sequence of partial sums \(S_{n},\) and therefore the series, converges to 1

Find a recursive definition for the sequence. $$1,5,14,30,55, \dots$$

Find the sum of the series. For what values of the variable does the series converge to this sum? $$2-4 z+8 z^{2}-16 z^{3}+\cdots$$

Use the alternating series test to decide whether the series converges. $$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{2 n+1}$$