Test the following infinite series for convergence or divergence.

• [math]\sum_{i=1}^\infty \frac{1}{7i-2}[/math]
• [math]\sum_{i=1}^\infty \frac{1}{7i^2-2}[/math]
• [math]\sum_{k=1}^\infty \frac{1}{\sqrt{k}}[/math]
• [math]\sum_{k=1}^\infty \frac{1}{\sqrt{k+7}}[/math]
• [math]\sum_{k=1}^\infty \frac{1}{k^{\frac32}}[/math]
• [math]\sum_{k=1}^\infty \frac{1}{\sqrt{k^2+1}}[/math]
• [math]\sum_{n=1}^\infty \frac{1}{\sqrt{n^3+2}}[/math]
• [math]\sum_{i=0}^\infty \frac{1}{1+i^2}[/math]
• [math]\sum_{i=4}^\infty \frac{1}{e^i}[/math]
• [math]\sum_{i=0}^\infty \frac{1}{i^2-3i+1}[/math].

Using the Integral Test for infinite series and the Comparison Test for integrals (Theorem \ref{thm 8.7.2}), determine whether each of the following series converges or diverges.

• [math]\sum_{k=1}^\infty e^{-k^2}[/math]
• [math]\sum_{i=1}^\infty \frac1{i^2} \sin \frac1{i^2}[/math].

Using the Integral Test, prove the theorem that, for positive [math]r[/math], the geometric series [math]\sum_{i=0}^\infty r^i[/math] converges if and only if [math]r \lt 1[/math].

Let [math]\sum_{i=m}^\infty a_i[/math] and [math]\sum_{i=m}^\infty b_i[/math] be two convergent nonnegative series. Using the Comparison Test, prove that the series [math]\sum_{i=m}^\infty a_ib_i[/math] also converges.

Prove the following theorem, which is hinted at in Example \ref{exam 9.3.3}. If [math]\sum_{i=m''^\infty b_i[/math] is a positive series (i.e., [math]b_i \gt 0[/math] for every integer [math]i \geq m[/math]) and if [math]\lim_{n\goesto\infty} \frac{a_n}{b_n} = 1[/math], then the series [math]\sum_{i=m}^\infty a_i[/math] converges if and only if [math]\sum_{i=m}^\infty b_i[/math] does.}