2) 

Remember the harmonic series is given by the sum  .  We know it diverges (it gets infinitely large), but there is simply no way to see this by just adding terms and seeing if the sequence of partial sums blows up or not.  To see what I mean, suppose we started with the first term of the harmonic series the day the universe was formed, 13 billion years ago, and added a new term every second.  About how large would the partial sum be today, assuming a 365-day year?  (Note: when you do this on Maple, it may give you a kooky-looking answer.  If it does, just use "evalf" to get a decimal.)