91Ó°ÊÓ

Use end behavior to compare the series to a \(p\) -series and predict whether the series converges or diverges. $$\sum_{n=1}^{\infty} \frac{1}{n^{4}+3 n^{3}+7}$$

Use end behavior to compare the series to a \(p\) -series and predict whether the series converges or diverges. $$\sum_{n=1}^{\infty} \frac{n-4}{\sqrt{n^{3}+n^{2}+8}}$$

Decide which of the following are geometric series. For those which are, give the first term and the ratio between successive terms. For those which are not, explain why not. $$5-10+20-40+80-\dots$$

Use comparison with \(\int_{1}^{\infty} x^{-3} d x\) to show that \(\sum_{n=2}^{\infty} 1 / n^{3}\) converges to a number less than or equal to \(1 / 2.\)

Find an expression for the general term of the series. Give the starting value of the index \((n \text { or } k,\) for example). $$(x-1)^{3}-\frac{(x-1)^{5}}{2 !}+\frac{(x-1)^{7}}{4 !}-\frac{(x-1)^{9}}{6 !}+\cdots$$

Find a formula for \(s_{n-1} n \geq 1\) $$1,3,7,15,31, \dots$$

Use the comparison test to determine whether the series converges. $$\sum_{n=1}^{\infty} \frac{n^{2}}{n^{4}+1}$$

Decide which of the following are geometric series. For those which are, give the first term and the ratio between successive terms. For those which are not, explain why not. $$2+1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots$$

Find an expression for the general term of the series. Give the starting value of the index \((n \text { or } k,\) for example). $$\frac{x-a}{1}+\frac{(x-a)^{2}}{2 \cdot 2 !}+\frac{(x-a)^{3}}{4 \cdot 3 !}+\frac{(x-a)^{4}}{8 \cdot 4 !}+\cdots$$

Use comparison with \(\int_{0}^{\infty} 1 /\left(x^{2}+1\right) d x\) to show that \(\sum_{n=1}^{\infty} 1 /\left(n^{2}+1\right)\) converges to a number less than or equal to \(\pi / 2.\)