Determine whether each of the following alternating series converges or diverges. Give the reasons for your answers.

• [a [math]\sum_{i=1}^\infty (-1)^i \frac{1}{\sqrt{i}}[/math]
• [math]\sum_{i=1}^\infty (-1)^i \frac{1}{i^2+1}[/math]
• [math]\sum_{k=1}^\infty (-1)^k \frac{k^2-1}{k^2+1}[/math]
• [math]\sum_{k=1}^\infty (-1)^k \frac{1}{(k^2+1)^{\frac13}}[/math]
• [math]\sum_{n=2}^\infty (-1)^n e^{-n}[/math]
• [math]\sum_{i=2}^\infty (-1)^i \frac{1}{\sqrt{2i^3-1}}[/math]
• [math]\sum_{i=0}^\infty \cos(i\pi)[/math]
• [math]\sum_{k=1}^\infty \frac{\cos(k\pi)}{k^2}[/math].

Prove that, for any infinite sequence [math]\{ a_n \}[/math] of real numbers, [math]\lim_{n\goesto\infty} a_n = 0[/math] if and only if [math]\lim_{n\goesto\infty} |a_n| = 0[/math]. (Hint: The proof is simple and straightforward. Go directly to the definition of convergence of an infinite sequence.)

For each of the series [math]\sum_{i=m}^\infty a_i[/math] in Problem Exercise, determine whether or not the corresponding series of absolute values [math]\sum_{i=m}^\infty |a_i|[/math] converges.

Give an example of an alternating series [math]\sum_{i=m}^\infty a_i[/math] which you can show converges, but which fails to satisfy condition (i) of the Convergence Test (\ref{thm 9.4.1i}).

The first of the following examples comes from the formula for a geometric series, and the last two follow from the theory developed later in this chapter:

• [math]\frac23 = \frac1{1+\frac12} = \sum_{i=0}^\infty (-\frac12)^i = 1 - \frac12 + \frac14 - \cdots[/math].
• [math]\ln 2 = \sum_{i=1}^\infty (-1)^{i+1} \frac1i = 1 - \frac12 + \frac13 - \frac14 + \cdots[/math].
• [math]\pi = 4 \arctan 1 = \sum_{i=0}^\infty (-1)^i \frac4{2i+1}= 4 - \frac43 + \frac45 - \frac47 + \cdots[/math].

If the value of each of these series is approximated by a partial sum [math]\sum_{i=m}^\infty a_i[/math], how large must [math]n[/math] be taken to ensure an error no greater than [math]0.1[/math], [math]0.01[/math], [math]0.001[/math], [math]10^{-6}[/math]?