Activity 8.4.7.
For (a)-(j), use appropriate tests to determine the convergence or divergence of the following series. Throughout, if a series is a convergent geometric series, find its sum.
(a)
\(\displaystyle\sum_{k=3}^{\infty} \ \frac{2}{\sqrt{k-2}}\)
(b)
\(\displaystyle\sum_{k=1}^{\infty} \ \frac{k}{1+2k}\)
(c)
\(\displaystyle\sum_{k=0}^{\infty} \ \frac{2k^2+1}{k^3+k+1}\)
(d)
\(\displaystyle\sum_{k=0}^{\infty} \ \frac{100^k}{k!}\)
(e)
\(\displaystyle\sum_{k=1}^{\infty} \ \frac{2^k}{5^k}\)
(f)
\(\displaystyle\sum_{k=1}^{\infty} \ \frac{k^3-1}{k^5+1}\)
(g)
\(\displaystyle\sum_{k=2}^{\infty} \ \frac{3^{k-1}}{7^k}\)
(h)
\(\displaystyle\sum_{k=2}^{\infty} \ \frac{1}{k^k}\)
(i)
\(\displaystyle\sum_{k=1}^{\infty} \ \frac{(-1)^{k+1}}{\sqrt{k+1}}\)
(j)
\(\displaystyle\sum_{k=2}^{\infty} \ \frac{1}{k \ln(k)}\)
(k)
Determine a value of \(n\) so that the \(n\)th partial sum \(S_n\) of the alternating series \(\displaystyle\sum_{n=2}^{\infty} \frac{(-1)^n}{\ln(n)}\) approximates the sum to within 0.001.
