DO: Test the following series for
convergence, divergence, conditional convergence and absolute
convergence, when possible. Hint: One way to proceed
with the first two series is by splitting the terms into three
parts. Remember, if $\sum a_n$ and $\sum b_n$ both converge,
then $\sum a_n+\sum
b_n=\sum(a_n+b_n)$ converges. (Similarly for three series.)
There are more examples to work on the next page, which you might
want to work on also before checking any answers.
The video below will discuss the preceding series. We have put
the written results below the video.
$\displaystyle\sum_n\left(2^{-n} + \frac{3n+5}{n^3}\right)$
converges as a sum of three
standard examples;
$\displaystyle\sum_n \left(2^{-n} + \frac{3n+5}{n^2}\right)$
diverges as a sum of three standard examples, two of which
converge and one of which diverges;
$\displaystyle\sum_n \left(2^{-n} + \frac{3n+5}{n}\right)$
diverges by the Divergence Test;
$\displaystyle\sum_n \frac{1}{n^2+n}$ converges by comparison to
$\displaystyle\sum \frac{1}{n^2}$;
$\displaystyle\sum_n \frac{n+\cos(n)}{n^3+n^2}$ also
converges by comparison to $\displaystyle\sum \frac{1}{n^2}$,
but we need the Limit Comparison Test;
$\displaystyle\sum_n \frac{1}{n \ln(n)}$ diverges by the Integral Test; and
$\displaystyle\sum_n \frac{1}{n \left(\ln(n)\right)^2}$
converges by the Integral Test.